Friable values of
polynomials
Friable values of polynomials
Greg Martin
Introduction
How often do the values of a polynomial
have only small prime factors?
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
Greg Martin
University of British Columbia
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
April 14, 2006
University of South Carolina Number Theory Seminar
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
notes to be placed on web page:
www.math.ubc.ca/∼gerg/talks.html
Shepherding the local factors
Sums of multiplicative functions
Summary
Outline
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
1
Introduction
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
2
Bounds for friable values of polynomials
How friable can values of general
polynomials be?
Can we have lots of friable values?
3
Conjecture for prime values of polynomials
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
4
Conjecture for friable values of polynomials
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Greg Martin
Introduction
1
Introduction
Friable integers
Friable numbers among values of polynomials
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
2
Bounds for friable values of polynomials
3
Conjecture for prime values of polynomials
4
Conjecture for friable values of polynomials
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable integers
Friable values of
polynomials
Greg Martin
Definition
Ψ(x, y ) is the number of integers up to x whose prime
factors are all at most y :
Ψ(x, y ) = #{n ≤ x : p | n =⇒ p ≤ y }
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Asymptotics: For a large range of x and y ,
log x
Ψ(x, y ) ∼ xρ
,
log y
where ρ(u) is the “Dickman–de Bruijn
rho-function”.
Interpretation: A “randomly chosen” integer of size X has
probability ρ(u) of being X 1/u -friable.
In this talk: Think of u = log x/ log y as being bounded
above, that is, y ≥ x ε for some ε > 0.
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable integers
Friable values of
polynomials
Greg Martin
Definition
Ψ(x, y ) is the number of integers up to x whose prime
factors are all at most y :
Ψ(x, y ) = #{n ≤ x : p | n =⇒ p ≤ y }
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Asymptotics: For a large range of x and y ,
log x
Ψ(x, y ) ∼ xρ
,
log y
where ρ(u) is the “Dickman–de Bruijn
rho-function”.
Interpretation: A “randomly chosen” integer of size X has
probability ρ(u) of being X 1/u -friable.
In this talk: Think of u = log x/ log y as being bounded
above, that is, y ≥ x ε for some ε > 0.
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable integers
Friable values of
polynomials
Greg Martin
Definition
Ψ(x, y ) is the number of integers up to x whose prime
factors are all at most y :
Ψ(x, y ) = #{n ≤ x : p | n =⇒ p ≤ y }
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Asymptotics: For a large range of x and y ,
log x
Ψ(x, y ) ∼ xρ
,
log y
where ρ(u) is the “Dickman–de Bruijn
rho-function”.
Interpretation: A “randomly chosen” integer of size X has
probability ρ(u) of being X 1/u -friable.
In this talk: Think of u = log x/ log y as being bounded
above, that is, y ≥ x ε for some ε > 0.
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The Dickman–de Bruijn ρ-function
Friable values of
polynomials
Greg Martin
Definition
Introduction
Friable integers
ρ(u) is the unique continuous solution of the
differential-difference equation uρ0 (u) = −ρ(u − 1) for u ≥ 1
that satisfies the initial condition ρ(u) = 1 for 0 ≤ u ≤ 1.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Example
Can we have lots of friable values?
Conjecture for prime
values of polynomials
For 1 ≤ u ≤ 2,
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
ρ0 (u) = −
ρ(u − 1)
1
= − =⇒ ρ(u) = C − log u.
u
u
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Since ρ(u) = 1, we have ρ(u) = 1 − log u for 1 ≤ u ≤ 2.
√
Consequence: Note that ρ(u) = 12 when u = e. Therefore
the “median
√ size” of the largest prime factor
of n is n1/ e .
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The Dickman–de Bruijn ρ-function
Friable values of
polynomials
Greg Martin
Definition
Introduction
Friable integers
ρ(u) is the unique continuous solution of the
differential-difference equation uρ0 (u) = −ρ(u − 1) for u ≥ 1
that satisfies the initial condition ρ(u) = 1 for 0 ≤ u ≤ 1.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Example
Can we have lots of friable values?
Conjecture for prime
values of polynomials
For 1 ≤ u ≤ 2,
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
ρ0 (u) = −
ρ(u − 1)
1
= − =⇒ ρ(u) = C − log u.
u
u
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Since ρ(u) = 1, we have ρ(u) = 1 − log u for 1 ≤ u ≤ 2.
√
Consequence: Note that ρ(u) = 12 when u = e. Therefore
the “median
√ size” of the largest prime factor
of n is n1/ e .
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The Dickman–de Bruijn ρ-function
Friable values of
polynomials
Greg Martin
Definition
Introduction
Friable integers
ρ(u) is the unique continuous solution of the
differential-difference equation uρ0 (u) = −ρ(u − 1) for u ≥ 1
that satisfies the initial condition ρ(u) = 1 for 0 ≤ u ≤ 1.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Example
Can we have lots of friable values?
Conjecture for prime
values of polynomials
For 1 ≤ u ≤ 2,
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
ρ0 (u) = −
ρ(u − 1)
1
= − =⇒ ρ(u) = C − log u.
u
u
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Since ρ(u) = 1, we have ρ(u) = 1 − log u for 1 ≤ u ≤ 2.
√
Consequence: Note that ρ(u) = 12 when u = e. Therefore
the “median
√ size” of the largest prime factor
of n is n1/ e .
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable numbers among values of
polynomials
Definition
Ψ(F ; x, y ) is the number of integers n up to x such that all
the prime factors of F (n) are all at most y :
Ψ(F ; x, y ) = #{1 ≤ n ≤ x : p | F (n) =⇒ p ≤ y }
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
When F (x) is a linear polynomial (friable numbers in
arithmetic progressions), we have
the same
log x asymptotic Ψ(F ; x, y ) ∼ ρ log
.
y
Knowing the size of Ψ(F ; x, y ) has applications to
analyzing the running time of modern factoring
algorithms (quadratic sieve, number field sieve).
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
A basic sort of question in number theory: are two
arithmetic properties (in this case, friability and being
the value of a polynomial) independent?
Summary
Friable numbers among values of
polynomials
Definition
Ψ(F ; x, y ) is the number of integers n up to x such that all
the prime factors of F (n) are all at most y :
Ψ(F ; x, y ) = #{1 ≤ n ≤ x : p | F (n) =⇒ p ≤ y }
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
When F (x) is a linear polynomial (friable numbers in
arithmetic progressions), we have
the same
log x asymptotic Ψ(F ; x, y ) ∼ ρ log
.
y
Knowing the size of Ψ(F ; x, y ) has applications to
analyzing the running time of modern factoring
algorithms (quadratic sieve, number field sieve).
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
A basic sort of question in number theory: are two
arithmetic properties (in this case, friability and being
the value of a polynomial) independent?
Summary
Friable numbers among values of
polynomials
Definition
Ψ(F ; x, y ) is the number of integers n up to x such that all
the prime factors of F (n) are all at most y :
Ψ(F ; x, y ) = #{1 ≤ n ≤ x : p | F (n) =⇒ p ≤ y }
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
When F (x) is a linear polynomial (friable numbers in
arithmetic progressions), we have
the same
log x asymptotic Ψ(F ; x, y ) ∼ ρ log
.
y
Knowing the size of Ψ(F ; x, y ) has applications to
analyzing the running time of modern factoring
algorithms (quadratic sieve, number field sieve).
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
A basic sort of question in number theory: are two
arithmetic properties (in this case, friability and being
the value of a polynomial) independent?
Summary
Friable values of
polynomials
Greg Martin
1
Introduction
Introduction
Friable integers
Friable values of polynomials
2
Bounds for friable values of polynomials
How friable can values of special polynomials be?
How friable can values of general polynomials be?
Can we have lots of friable values?
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
3
Conjecture for prime values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
4
Conjecture for friable values of polynomials
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
How friable can values of special
polynomials be?
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
For binomials, there’s a nice trick which yields:
Theorem (Schinzel, 1967)
For any nonzero integers A and B, any positive integer d,
and any ε > 0, there are infinitely many numbers n for
which And + B is nε -friable.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Balog and Wooley (1998), building on an idea of
Eggleton and Selfridge, extended this result to
products of binomials
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
L
Y
(Aj ndj + Bj ).
j=1
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
How friable can values of special
polynomials be?
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
For binomials, there’s a nice trick which yields:
Theorem (Schinzel, 1967)
For any nonzero integers A and B, any positive integer d,
and any ε > 0, there are infinitely many numbers n for
which And + B is nε -friable.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Balog and Wooley (1998), building on an idea of
Eggleton and Selfridge, extended this result to
products of binomials
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
L
Y
(Aj ndj + Bj ).
j=1
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Proof for an explicit binomial
Friable values of
polynomials
Greg Martin
Example
Introduction
For any ε > 0, there are infinitely many numbers n for which
F (n) = 3n5 + 7 is nε -friable.
Define nk = 38k −1 72k . Then
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
F (nk ) = 35(8k −1)+1 75(2k ) + 7 = −7 (−34 7)10k −1 − 1
factors into values of cyclotomic polynomials:
Y
Φm (−34 7).
F (nk ) = −7
m|(10k −1)
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Φm (x) =
Y
x − e2πir /m
1≤r ≤m
(r ,m)=1
Φm has integer coefficients and degree φ(m)
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Proof for an explicit binomial
Friable values of
polynomials
Greg Martin
Example
Introduction
For any ε > 0, there are infinitely many numbers n for which
F (n) = 3n5 + 7 is nε -friable.
Define nk = 38k −1 72k . Then
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
F (nk ) = 35(8k −1)+1 75(2k ) + 7 = −7 (−34 7)10k −1 − 1
factors into values of cyclotomic polynomials:
Y
Φm (−34 7).
F (nk ) = −7
m|(10k −1)
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Φm (x) =
Y
x − e2πir /m
1≤r ≤m
(r ,m)=1
Φm has integer coefficients and degree φ(m)
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Proof for an explicit binomial
Friable values of
polynomials
Greg Martin
Example
Introduction
For any ε > 0, there are infinitely many numbers n for which
F (n) = 3n5 + 7 is nε -friable.
Define nk = 38k −1 72k . Then
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
F (nk ) = 35(8k −1)+1 75(2k ) + 7 = −7 (−34 7)10k −1 − 1
factors into values of cyclotomic polynomials:
Y
Φm (−34 7).
F (nk ) = −7
m|(10k −1)
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Φm (x) =
Y
x − e2πir /m
1≤r ≤m
(r ,m)=1
Φm has integer coefficients and degree φ(m)
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Proof for an explicit binomial
Friable values of
polynomials
Greg Martin
Example
Introduction
For any ε > 0, there are infinitely many numbers n for which
F (n) = 3n5 + 7 is nε -friable.
Define nk = 38k −1 72k . Then
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
F (nk ) = 35(8k −1)+1 75(2k ) + 7 = −7 (−34 7)10k −1 − 1
factors into values of cyclotomic polynomials:
Y
Φm (−34 7).
F (nk ) = −7
m|(10k −1)
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Φm (x) =
Y
x − e2πir /m
1≤r ≤m
(r ,m)=1
Φm has integer coefficients and degree φ(m)
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Proof for an explicit binomial
Friable values of
polynomials
Greg Martin
Example
Introduction
For any ε > 0, there are infinitely many numbers n for which
F (n) = 3n5 + 7 is nε -friable.
Define nk = 38k −1 72k . Then
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
F (nk ) = 35(8k −1)+1 75(2k ) + 7 = −7 (−34 7)10k −1 − 1
factors into values of cyclotomic polynomials:
Y
Φm (−34 7).
F (nk ) = −7
m|(10k −1)
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Φm (x) =
Y
x − e2πir /m
1≤r ≤m
(r ,m)=1
Φm has integer coefficients and degree φ(m)
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Greg Martin
From the last slide
Introduction
F (n) = 3n5 + 7
Y
F (nk ) = −7
Φm (−34 7)
m|(10k −1)
nk = 38k −1 72k
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
primes dividing F (nk ) are ≤
max
m|(10k −1)
Φm (−34 7)
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
4k
nk is roughly (3 7) , but the largest prime factor of
F (nk ) is bounded by roughly (34 7)φ(10k −1)
infinitely many k with φ(10k − 1)/4k < ε
How many such friable values? F ,ε log x, for n ≤ x
ε can be made quantitative n
cF / log log log n
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Φm (x) is roughly x φ(m) ≤ x φ(10k −1)
4
How friable can values of general
polynomials be?
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
-friable values
Summary
Friable values of
polynomials
Greg Martin
From the last slide
Introduction
F (n) = 3n5 + 7
Y
F (nk ) = −7
Φm (−34 7)
m|(10k −1)
nk = 38k −1 72k
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
primes dividing F (nk ) are ≤
max
m|(10k −1)
Φm (−34 7)
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
4k
nk is roughly (3 7) , but the largest prime factor of
F (nk ) is bounded by roughly (34 7)φ(10k −1)
infinitely many k with φ(10k − 1)/4k < ε
How many such friable values? F ,ε log x, for n ≤ x
ε can be made quantitative n
cF / log log log n
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Φm (x) is roughly x φ(m) ≤ x φ(10k −1)
4
How friable can values of general
polynomials be?
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
-friable values
Summary
Friable values of
polynomials
Greg Martin
From the last slide
Introduction
F (n) = 3n5 + 7
Y
F (nk ) = −7
Φm (−34 7)
m|(10k −1)
nk = 38k −1 72k
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
primes dividing F (nk ) are ≤
max
m|(10k −1)
Φm (−34 7)
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
4k
nk is roughly (3 7) , but the largest prime factor of
F (nk ) is bounded by roughly (34 7)φ(10k −1)
infinitely many k with φ(10k − 1)/4k < ε
How many such friable values? F ,ε log x, for n ≤ x
ε can be made quantitative n
cF / log log log n
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Φm (x) is roughly x φ(m) ≤ x φ(10k −1)
4
How friable can values of general
polynomials be?
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
-friable values
Summary
Friable values of
polynomials
Greg Martin
From the last slide
Introduction
F (n) = 3n5 + 7
Y
F (nk ) = −7
Φm (−34 7)
m|(10k −1)
nk = 38k −1 72k
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
primes dividing F (nk ) are ≤
max
m|(10k −1)
Φm (−34 7)
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
4k
nk is roughly (3 7) , but the largest prime factor of
F (nk ) is bounded by roughly (34 7)φ(10k −1)
infinitely many k with φ(10k − 1)/4k < ε
How many such friable values? F ,ε log x, for n ≤ x
ε can be made quantitative n
cF / log log log n
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Φm (x) is roughly x φ(m) ≤ x φ(10k −1)
4
How friable can values of general
polynomials be?
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
-friable values
Summary
Polynomial factorizations
Friable values of
polynomials
Greg Martin
Example
Introduction
The polynomial F (x + F (x)) is always divisible by F (x). In
2
particular, if deg F = d, then F (x + F (x)) is roughly x d yet
2
is automatically roughly x d −d -friable.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Mnemonic
Can we have lots of friable values?
x + F (x) ≡ x (mod F (x))
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Special case:
A uniform version of Hypothesis H
If F (x) is quadratic with lead coefficient a, then
F (x + F (x)) = F (x) · aF x + a1 .
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
In particular, if F (x) = x 2 + bx + c, then
Sums of multiplicative functions
Summary
F (x + F (x)) = F (x)F (x + 1).
Polynomial factorizations
Friable values of
polynomials
Greg Martin
Example
Introduction
The polynomial F (x + F (x)) is always divisible by F (x). In
2
particular, if deg F = d, then F (x + F (x)) is roughly x d yet
2
is automatically roughly x d −d -friable.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Mnemonic
Can we have lots of friable values?
x + F (x) ≡ x (mod F (x))
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Special case:
A uniform version of Hypothesis H
If F (x) is quadratic with lead coefficient a, then
F (x + F (x)) = F (x) · aF x + a1 .
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
In particular, if F (x) = x 2 + bx + c, then
Sums of multiplicative functions
Summary
F (x + F (x)) = F (x)F (x + 1).
Friable values of
polynomials
A refinement of Schinzel
Greg Martin
Idea: use the reciprocal polynomial x d F (1/x).
Introduction
Friable integers
d
Restrict to F (x) = x + a2 x
d−2
+ . . . for simplicity.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
Proposition
How friable can values of general
polynomials be?
Let h(x) be a polynomial such that xh(x) − 1 is divisible by
x d F (1/x). Then F (h(x)) is divisible by x d F (1/x). In
particular, we can take deg h = d − 1, in which case F (h(x))
2
2
is roughly x d −d yet is automatically roughly x d −2d -friable.
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Mnemonic
Statement of the conjecture
h(x) ≡ 1/x (mod F (1/x))
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Note: The proposition isn’t true for d = 2, since the leftover
“factor” of degree 22 − 2 · 2 = 0 is a constant.
Summary
Friable values of
polynomials
A refinement of Schinzel
Greg Martin
Idea: use the reciprocal polynomial x d F (1/x).
Introduction
Friable integers
d
Restrict to F (x) = x + a2 x
d−2
+ . . . for simplicity.
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
Proposition
How friable can values of general
polynomials be?
Let h(x) be a polynomial such that xh(x) − 1 is divisible by
x d F (1/x). Then F (h(x)) is divisible by x d F (1/x). In
particular, we can take deg h = d − 1, in which case F (h(x))
2
2
is roughly x d −d yet is automatically roughly x d −2d -friable.
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Mnemonic
Statement of the conjecture
h(x) ≡ 1/x (mod F (1/x))
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Note: The proposition isn’t true for d = 2, since the leftover
“factor” of degree 22 − 2 · 2 = 0 is a constant.
Summary
Recursively use Schinzel’s construction
Friable values of
polynomials
Greg Martin
Dm : an unspecified polynomial of degree m
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
Example
deg F (x) = 4. Use Schinzel’s construction repeatedly:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
D12 = F (D3 ) = D4 D8
D84 = F (D21 ) = D28 D8 D48
D3984 = F (D987 ) = D1316 D376 D48 D2208
“score” = 8/3
“score” = 16/7
“score” = 736/329
For deg F = 2, begin with F (D4 ) = D2 D2 D4 .
Specifically,
F x + F (x) + F x + F (x) = F (x) · aF x + a1 · D4 .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For deg F = 3, begin with F (D4 ) = D3 D3 D6 .
Recursively use Schinzel’s construction
Friable values of
polynomials
Greg Martin
Dm : an unspecified polynomial of degree m
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
Example
deg F (x) = 4. Use Schinzel’s construction repeatedly:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
D12 = F (D3 ) = D4 D8
D84 = F (D21 ) = D28 D8 D48
D3984 = F (D987 ) = D1316 D376 D48 D2208
“score” = 8/3
“score” = 16/7
“score” = 736/329
For deg F = 2, begin with F (D4 ) = D2 D2 D4 .
Specifically,
F x + F (x) + F x + F (x) = F (x) · aF x + a1 · D4 .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For deg F = 3, begin with F (D4 ) = D3 D3 D6 .
Recursively use Schinzel’s construction
Friable values of
polynomials
Greg Martin
Dm : an unspecified polynomial of degree m
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
Example
deg F (x) = 4. Use Schinzel’s construction repeatedly:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
D12 = F (D3 ) = D4 D8
D84 = F (D21 ) = D28 D8 D48
D3984 = F (D987 ) = D1316 D376 D48 D2208
“score” = 8/3
“score” = 16/7
“score” = 736/329
For deg F = 2, begin with F (D4 ) = D2 D2 D4 .
Specifically,
F x + F (x) + F x + F (x) = F (x) · aF x + a1 · D4 .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For deg F = 3, begin with F (D4 ) = D3 D3 D6 .
Recursively use Schinzel’s construction
Friable values of
polynomials
Greg Martin
Dm : an unspecified polynomial of degree m
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
Example
deg F (x) = 4. Use Schinzel’s construction repeatedly:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
D12 = F (D3 ) = D4 D8
D84 = F (D21 ) = D28 D8 D48
D3984 = F (D987 ) = D1316 D376 D48 D2208
“score” = 8/3
“score” = 16/7
“score” = 736/329
For deg F = 2, begin with F (D4 ) = D2 D2 D4 .
Specifically,
F x + F (x) + F x + F (x) = F (x) · aF x + a1 · D4 .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For deg F = 3, begin with F (D4 ) = D3 D3 D6 .
Recursively use Schinzel’s construction
Friable values of
polynomials
Greg Martin
Dm : an unspecified polynomial of degree m
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
Example
deg F (x) = 4. Use Schinzel’s construction repeatedly:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
D12 = F (D3 ) = D4 D8
D84 = F (D21 ) = D28 D8 D48
D3984 = F (D987 ) = D1316 D376 D48 D2208
“score” = 8/3
“score” = 16/7
“score” = 736/329
For deg F = 2, begin with F (D4 ) = D2 D2 D4 .
Specifically,
F x + F (x) + F x + F (x) = F (x) · aF x + a1 · D4 .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For deg F = 3, begin with F (D4 ) = D3 D3 D6 .
How friable can values of general
polynomials be?
d ≥ 4: define s(d) = d
∞ Y
j=1
Greg Martin
Introduction
1
1−
, where
uj (d)
u1 (d) = d − 1 and uj+1 (d) = uj (d)2 − 2
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
s(2) = s(4)/4 and s(3) = s(6)/4
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Theorem
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
(Schinzel, 1967) Given a polynomial F (x) of degree d ≥ 2,
there are infinitely many numbers n for which F (n) is
ns(d) -friable.
F (n)
degree 1
degree 2
degree 3
degree 4
Friable values of
polynomials
can be n? -friable
ε
0.55902
1.14564
2.23607
F (n)
degree 5
degree 6
degree 7
degree d
can be n? -friable
3.46410
4.58258
5.65685
≈ d − 1 − 2/d
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
How friable can values of general
polynomials be?
d ≥ 4: define s(d) = d
∞ Y
j=1
Greg Martin
Introduction
1
1−
, where
uj (d)
u1 (d) = d − 1 and uj+1 (d) = uj (d)2 − 2
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
s(2) = s(4)/4 and s(3) = s(6)/4
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Theorem
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
(Schinzel, 1967) Given a polynomial F (x) of degree d ≥ 2,
there are infinitely many numbers n for which F (n) is
ns(d) -friable.
F (n)
degree 1
degree 2
degree 3
degree 4
Friable values of
polynomials
can be n? -friable
ε
0.55902
1.14564
2.23607
F (n)
degree 5
degree 6
degree 7
degree d
can be n? -friable
3.46410
4.58258
5.65685
≈ d − 1 − 2/d
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
How friable can values of general
polynomials be?
d ≥ 4: define s(d) = d
∞ Y
j=1
Greg Martin
Introduction
1
1−
, where
uj (d)
u1 (d) = d − 1 and uj+1 (d) = uj (d)2 − 2
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
s(2) = s(4)/4 and s(3) = s(6)/4
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Theorem
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
(Schinzel, 1967) Given a polynomial F (x) of degree d ≥ 2,
there are infinitely many numbers n for which F (n) is
ns(d) -friable.
F (n)
degree 1
degree 2
degree 3
degree 4
Friable values of
polynomials
can be n? -friable
ε
0.55902
1.14564
2.23607
F (n)
degree 5
degree 6
degree 7
degree d
can be n? -friable
3.46410
4.58258
5.65685
≈ d − 1 − 2/d
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
How friable can values of general
polynomials be?
d ≥ 4: define s(d) = d
∞ Y
j=1
Greg Martin
Introduction
1
1−
, where
uj (d)
u1 (d) = d − 1 and uj+1 (d) = uj (d)2 − 2
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
s(2) = s(4)/4 and s(3) = s(6)/4
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Theorem
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
(Schinzel, 1967) Given a polynomial F (x) of degree d ≥ 2,
there are infinitely many numbers n for which F (n) is
ns(d) -friable.
F (n)
degree 1
degree 2
degree 3
degree 4
Friable values of
polynomials
can be n? -friable
ε
0.55902
1.14564
2.23607
F (n)
degree 5
degree 6
degree 7
degree d
can be n? -friable
3.46410
4.58258
5.65685
≈ d − 1 − 2/d
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Polynomial substitution yields small lower
bounds
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Special case
Friable values of polynomials
Given a quadratic polynomial F (x), there are infinitely many
numbers n for which F (n) is n0.55902 -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Example
Conjecture for prime
values of polynomials
To obtain n for which F (n) is n0.56 -friable:
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
D168 = F (D84 ) = D42 D42 D28 D8 D48
“score” = 4/7 > 0.56
D7896 = F (D3948 )
“score” = 92/329
= D1974 D1974 D1316 D376 D48 D2208
< 0.56
The counting function of such n is about x
1/3948
.
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
“Improvement” Balog, M., Wooley can get x 2/3948 and an
analogous improvement for deg F = 3.
Summary
Polynomial substitution yields small lower
bounds
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Special case
Friable values of polynomials
Given a quadratic polynomial F (x), there are infinitely many
numbers n for which F (n) is n0.55902 -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Example
Conjecture for prime
values of polynomials
To obtain n for which F (n) is n0.56 -friable:
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
“score” = 4/7 > 0.56
D168 = F (D84 ) = D42 D42 D28 D8 D48
“score” = 92/329
D7896 = F (D3948 )
< 0.56
= D1974 D1974 D1316 D376 D48 D2208
The counting function of such n is about x
1/3948
.
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
“Improvement” Balog, M., Wooley can get x 2/3948 and an
analogous improvement for deg F = 3.
Summary
Polynomial substitution yields small lower
bounds
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Special case
Friable values of polynomials
Given a quadratic polynomial F (x), there are infinitely many
numbers n for which F (n) is n0.55902 -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Example
Conjecture for prime
values of polynomials
To obtain n for which F (n) is n0.56 -friable:
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
“score” = 4/7 > 0.56
D168 = F (D84 ) = D42 D42 D28 D8 D48
“score” = 92/329
D7896 = F (D3948 )
< 0.56
= D1974 D1974 D1316 D376 D48 D2208
The counting function of such n is about x
1/3948
.
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
“Improvement” Balog, M., Wooley can get x 2/3948 and an
analogous improvement for deg F = 3.
Summary
Polynomial substitution yields small lower
bounds
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Special case
Friable values of polynomials
Given a quadratic polynomial F (x), there are infinitely many
numbers n for which F (n) is n0.55902 -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Example
Conjecture for prime
values of polynomials
To obtain n for which F (n) is n0.56 -friable:
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
“score” = 4/7 > 0.56
D168 = F (D84 ) = D42 D42 D28 D8 D48
“score” = 92/329
D7896 = F (D3948 )
< 0.56
= D1974 D1974 D1316 D376 D48 D2208
The counting function of such n is about x
1/3948
.
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
“Improvement” Balog, M., Wooley can get x 2/3948 and an
analogous improvement for deg F = 3.
Summary
Can we have lots of friable values?
Friable values of
polynomials
Greg Martin
Introduction
Our expectation
Friable integers
Friable values of polynomials
For any ε > 0, a positive proportion of values F (n) are
nε -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
We know this for:
How friable can values of general
polynomials be?
Can we have lots of friable values?
linear polynomials (arithmetic progressions)
Hildebrand, then Balog and Ruzsa: F (n) = n(an + b),
values nε -friable for any ε > 0
Hildebrand: F (n) = (n + 1) · · · (n + L), values
nβ -friable for any β > e−1/(L−1)
Note: ρ(e
−1/L
)=1−
1
L,
so β > e
−1/L
is trivial
Dartyge: F (n) = n2 + 1, values nβ -friable for any
β > 149/179
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Can we have lots of friable values?
Friable values of
polynomials
Greg Martin
Introduction
Our expectation
Friable integers
Friable values of polynomials
For any ε > 0, a positive proportion of values F (n) are
nε -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
We know this for:
How friable can values of general
polynomials be?
Can we have lots of friable values?
linear polynomials (arithmetic progressions)
Hildebrand, then Balog and Ruzsa: F (n) = n(an + b),
values nε -friable for any ε > 0
Hildebrand: F (n) = (n + 1) · · · (n + L), values
nβ -friable for any β > e−1/(L−1)
Note: ρ(e
−1/L
)=1−
1
L,
so β > e
−1/L
is trivial
Dartyge: F (n) = n2 + 1, values nβ -friable for any
β > 149/179
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Can we have lots of friable values?
Friable values of
polynomials
Greg Martin
Introduction
Our expectation
Friable integers
Friable values of polynomials
For any ε > 0, a positive proportion of values F (n) are
nε -friable.
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
We know this for:
How friable can values of general
polynomials be?
Can we have lots of friable values?
linear polynomials (arithmetic progressions)
Hildebrand, then Balog and Ruzsa: F (n) = n(an + b),
values nε -friable for any ε > 0
Hildebrand: F (n) = (n + 1) · · · (n + L), values
nβ -friable for any β > e−1/(L−1)
Note: ρ(e
−1/L
)=1−
1
L,
so β > e
−1/L
is trivial
Dartyge: F (n) = n2 + 1, values nβ -friable for any
β > 149/179
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Theorem (Dartyge, M., Tenenbaum, 2001)
Let F (x) be any polynomial, let d be the highest degree of
any irreducible factor of F , and let F have exactly K distinct
irreducible factors of degree d. Then for any ε > 0, a
positive proportion of values F (n) are nd−1/K +ε -friable.
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
d−1
or higher, only irreducible
Remark: for friability of level n
factors of degree ≥ d matter
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Trivial: nd -friable
A uniform version of Hypothesis H
Can remove the ε at the cost of the counting function: the
number of n ≤ x for which F (n) is nd−1/K -friable is
x
.
(log x)K (log 4−1+ε)
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Theorem (Dartyge, M., Tenenbaum, 2001)
Let F (x) be any polynomial, let d be the highest degree of
any irreducible factor of F , and let F have exactly K distinct
irreducible factors of degree d. Then for any ε > 0, a
positive proportion of values F (n) are nd−1/K +ε -friable.
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
d−1
or higher, only irreducible
Remark: for friability of level n
factors of degree ≥ d matter
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Trivial: nd -friable
A uniform version of Hypothesis H
Can remove the ε at the cost of the counting function: the
number of n ≤ x for which F (n) is nd−1/K -friable is
x
.
(log x)K (log 4−1+ε)
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Greg Martin
Introduction
1
2
3
Introduction
Bounds for friable values of polynomials
Conjecture for prime values of polynomials
Schinzel’s “Hypothesis H” (Bateman–Horn conjecture)
A uniform version of Hypothesis H
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
4
Conjecture for friable values of polynomials
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Schinzel’s “Hypothesis H” (Bateman–Horn
conjecture)
Friable values of
polynomials
Greg Martin
Introduction
Definition
Friable integers
Friable values of polynomials
π(F ; x) = #{n ≤ x :
f (n) is prime for each irreducible factor f of F }
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Conjecture: π(F ; x) is asymptotic to H(F ) li(F ; x), where:
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Z
dt
.
log |F1 (t)| . . . log |FL (t)|
li(F ; x) =
0<t<x
min{|F1 (t)|,...,|FL (t)|}≥2
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
H(F ) =
Y
p
1
1−
p
−L σ(F ; p)
1−
.
p
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
L: the number of distinct irreducible factors of F
σ(F ; n): the number of solutions of F (a) ≡ 0 (mod n)
Summary
Schinzel’s “Hypothesis H” (Bateman–Horn
conjecture)
Friable values of
polynomials
Greg Martin
Introduction
Definition
Friable integers
Friable values of polynomials
π(F ; x) = #{n ≤ x :
f (n) is prime for each irreducible factor f of F }
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Conjecture: π(F ; x) is asymptotic to H(F ) li(F ; x), where:
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Z
dt
.
log |F1 (t)| . . . log |FL (t)|
li(F ; x) =
0<t<x
min{|F1 (t)|,...,|FL (t)|}≥2
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
H(F ) =
Y
p
1
1−
p
−L σ(F ; p)
1−
.
p
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
L: the number of distinct irreducible factors of F
σ(F ; n): the number of solutions of F (a) ≡ 0 (mod n)
Summary
Schinzel’s “Hypothesis H” (Bateman–Horn
conjecture)
Friable values of
polynomials
Greg Martin
Introduction
Definition
Friable integers
Friable values of polynomials
π(F ; x) = #{n ≤ x :
f (n) is prime for each irreducible factor f of F }
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Conjecture: π(F ; x) is asymptotic to H(F ) li(F ; x), where:
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Z
dt
.
log |F1 (t)| . . . log |FL (t)|
li(F ; x) =
0<t<x
min{|F1 (t)|,...,|FL (t)|}≥2
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
H(F ) =
Y
p
1
1−
p
−L σ(F ; p)
1−
.
p
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
L: the number of distinct irreducible factors of F
σ(F ; n): the number of solutions of F (a) ≡ 0 (mod n)
Summary
Schinzel’s “Hypothesis H” (Bateman–Horn
conjecture)
Friable values of
polynomials
Greg Martin
Introduction
Definition
Friable integers
Friable values of polynomials
π(F ; x) = #{n ≤ x :
f (n) is prime for each irreducible factor f of F }
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Conjecture: π(F ; x) is asymptotic to H(F ) li(F ; x), where:
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Z
dt
.
log |F1 (t)| . . . log |FL (t)|
li(F ; x) =
0<t<x
min{|F1 (t)|,...,|FL (t)|}≥2
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
H(F ) =
Y
p
1
1−
p
−L σ(F ; p)
1−
.
p
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
L: the number of distinct irreducible factors of F
σ(F ; n): the number of solutions of F (a) ≡ 0 (mod n)
Summary
A uniform version of Hypothesis H
Friable values of
polynomials
Greg Martin
Hypothesis UH
Introduction
Friable integers
π(F ; t) − H(F ) li(F ; t) d,B
H(F )t
1+
(log t)L+1
uniformly for all polynomials F of degree d with L distinct
irreducible factors, each of which has coefficients bounded
by t B in absolute value.
t
li(F ; t) is asymptotic to
for fixed F
(log t)L
For d = K = 1, equivalent to expected number of
primes, in an interval of length y = x ε near x, in an
arithmetic progression to a modulus q ≤ y 1−ε
Don’t really need this strong a uniformity, but rather on
average over some funny family to be described later
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
A uniform version of Hypothesis H
Friable values of
polynomials
Greg Martin
Hypothesis UH
Introduction
Friable integers
π(F ; t) − H(F ) li(F ; t) d,B
H(F )t
1+
(log t)L+1
uniformly for all polynomials F of degree d with L distinct
irreducible factors, each of which has coefficients bounded
by t B in absolute value.
t
li(F ; t) is asymptotic to
for fixed F
(log t)L
For d = K = 1, equivalent to expected number of
primes, in an interval of length y = x ε near x, in an
arithmetic progression to a modulus q ≤ y 1−ε
Don’t really need this strong a uniformity, but rather on
average over some funny family to be described later
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Greg Martin
1
Introduction
Introduction
Friable integers
Friable values of polynomials
2
Bounds for friable values of polynomials
3
Conjecture for prime values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
4
Conjecture for friable values of polynomials
Statement of the conjecture
Reduction to convenient polynomials
Translation into prime values of polynomials
Shepherding the local factors
Sums of multiplicative functions
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
What would we expect on probablistic
grounds?
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Let F (x) = f1 (x) · · · fL (x), where deg fj (x) = dj . Let u > 0.
fj (n) is roughly ndj , and integers of that size are
n1/u -friable with probability ρ(dj u).
Are the friabilities of the various factors fj (n)
independent? This would lead to a prediction involving
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
x
L
Y
A uniform version of Hypothesis H
ρ(dj u).
j=1
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
What about local densities depending on the arithmetic
of F (as in Hypothesis H)?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
What would we expect on probablistic
grounds?
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Let F (x) = f1 (x) · · · fL (x), where deg fj (x) = dj . Let u > 0.
fj (n) is roughly ndj , and integers of that size are
n1/u -friable with probability ρ(dj u).
Are the friabilities of the various factors fj (n)
independent? This would lead to a prediction involving
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
x
L
Y
A uniform version of Hypothesis H
ρ(dj u).
j=1
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
What about local densities depending on the arithmetic
of F (as in Hypothesis H)?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
What would we expect on probablistic
grounds?
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Let F (x) = f1 (x) · · · fL (x), where deg fj (x) = dj . Let u > 0.
fj (n) is roughly ndj , and integers of that size are
n1/u -friable with probability ρ(dj u).
Are the friabilities of the various factors fj (n)
independent? This would lead to a prediction involving
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
x
L
Y
A uniform version of Hypothesis H
ρ(dj u).
j=1
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
What about local densities depending on the arithmetic
of F (as in Hypothesis H)?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Conjecture for friable values of polynomials
Friable values of
polynomials
Greg Martin
Conjecture
Introduction
Let F (x) be any polynomial, let f1 , . . . , fL be its distinct
irreducible factors, and let d1 , . . . , dL be their degrees. Then
Ψ(F ; x, x
1/u
)=x
L
Y
j=1
x
ρ(dj u) + O
log x
for all 0 < u.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
If F irreducible: Ψ(F ; x, x 1/u ) = xρ(du) + O(x/ log x) for
0 < u.
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Remark: Rather more controversial than
Hypothesis H.
Sums of multiplicative functions
Summary
Conjecture for friable values of polynomials
Friable values of
polynomials
Greg Martin
Theorem (M., 2002)
Introduction
Assume Hypothesis UH. Let F (x) be any polynomial, let
f1 , . . . , fL be its distinct irreducible factors, and let d1 , . . . , dL
be their degrees. Let d = max{d1 , . . . , dL }, and let F have
exactly K distinct irreducible factors of degree d. Then
Ψ(F ; x, x
1/u
)=x
L
Y
j=1
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
x
ρ(dj u) + O
log x
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
for all 0 < u < 1/(d − 1/K ).
Conjecture for friable
values of polynomials
Statement of the conjecture
If F irreducible: Ψ(F ; x, x 1/u ) = xρ(du) + O(x/ log x) for
0 < u < 1/(d − 1).
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Trivial: 0 < u < 1/d.
Reason to talk about more general K : There is one part of
the argument that causes an additional
difficulty when K > 1.
Sums of multiplicative functions
Summary
Conjecture for friable values of polynomials
Friable values of
polynomials
Greg Martin
Theorem (M., 2002)
Introduction
Assume Hypothesis UH. Let F (x) be any polynomial, let
f1 , . . . , fL be its distinct irreducible factors, and let d1 , . . . , dL
be their degrees. Let d = max{d1 , . . . , dL }, and let F have
exactly K distinct irreducible factors of degree d. Then
Ψ(F ; x, x
1/u
)=x
L
Y
j=1
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
x
ρ(dj u) + O
log x
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
for all 0 < u < 1/(d − 1/K ).
Conjecture for friable
values of polynomials
Statement of the conjecture
If F irreducible: Ψ(F ; x, x 1/u ) = xρ(du) + O(x/ log x) for
0 < u < 1/(d − 1).
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Trivial: 0 < u < 1/d.
Reason to talk about more general K : There is one part of
the argument that causes an additional
difficulty when K > 1.
Sums of multiplicative functions
Summary
Reduction to convenient polynomials
Friable values of
polynomials
Greg Martin
Without loss of generality, we may assume:
Introduction
1
2
F (x) is the product of distinct irreducible polynomials
f1 (x), . . . , fK (x), all of the same degree d.
F (x) takes at least one nonzero value modulo every
prime.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
3
No two distinct irreducible factors fi (x), fj (x) of F (x)
have a common zero modulo any prime.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
(1) is acceptable since the friability level exceeds x d−1 .
(2) is not a necessary condition to have friable values
(as it is to have prime values). Nevertheless, we can
still reduce to this case.
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Both (2) and (3) are achieved by looking at values of
F (x) on suitable arithmetic progressions F (Qx + R)
separately.
Summary
Reduction to convenient polynomials
Friable values of
polynomials
Greg Martin
Without loss of generality, we may assume:
Introduction
1
2
F (x) is the product of distinct irreducible polynomials
f1 (x), . . . , fK (x), all of the same degree d.
F (x) takes at least one nonzero value modulo every
prime.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
3
No two distinct irreducible factors fi (x), fj (x) of F (x)
have a common zero modulo any prime.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
(1) is acceptable since the friability level exceeds x d−1 .
(2) is not a necessary condition to have friable values
(as it is to have prime values). Nevertheless, we can
still reduce to this case.
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Both (2) and (3) are achieved by looking at values of
F (x) on suitable arithmetic progressions F (Qx + R)
separately.
Summary
Reduction to convenient polynomials
Friable values of
polynomials
Greg Martin
Without loss of generality, we may assume:
Introduction
1
2
F (x) is the product of distinct irreducible polynomials
f1 (x), . . . , fK (x), all of the same degree d.
F (x) takes at least one nonzero value modulo every
prime.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
3
No two distinct irreducible factors fi (x), fj (x) of F (x)
have a common zero modulo any prime.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
(1) is acceptable since the friability level exceeds x d−1 .
(2) is not a necessary condition to have friable values
(as it is to have prime values). Nevertheless, we can
still reduce to this case.
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Both (2) and (3) are achieved by looking at values of
F (x) on suitable arithmetic progressions F (Qx + R)
separately.
Summary
Reduction to convenient polynomials
Friable values of
polynomials
Greg Martin
Without loss of generality, we may assume:
Introduction
1
2
F (x) is the product of distinct irreducible polynomials
f1 (x), . . . , fK (x), all of the same degree d.
F (x) takes at least one nonzero value modulo every
prime.
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
3
No two distinct irreducible factors fi (x), fj (x) of F (x)
have a common zero modulo any prime.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Under (1), we want to prove that
Statement of the conjecture
Ψ(F ; x, x
1/u
x
) = xρ(du) + O
log x
K
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
for all 0 < u < 1/(d − 1/K ).
Summary
Inclusion-exclusion on irreducible factors
Friable values of
polynomials
Greg Martin
Proposition
Introduction
Let F be a primitive polynomial, and let F1 , . . . , FK denote
the distinct irreducible factors of F . Then for x ≥ y ≥ 1,
X
X
Ψ(F ; x, y ) = bxc+
(−1)k
M(Fi1 . . . Fik ; x, y ).
1≤k ≤K
1≤i1 <···<ik ≤K
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Definition
M(f ; x, y ) = #{1 ≤ n ≤ x : for each irreducible factor g of f ,
there exists a prime p > y such that p | g(n)}.
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Inclusion-exclusion on irreducible factors
Friable values of
polynomials
Greg Martin
Proposition
Introduction
Let F be a primitive polynomial, and let F1 , . . . , FK denote
the distinct irreducible factors of F . Then for x ≥ y ≥ 1,
X
X
Ψ(F ; x, y ) = bxc+
(−1)k
M(Fi1 . . . Fik ; x, y ).
1≤k ≤K
1≤i1 <···<ik ≤K
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Definition
M(f ; x, y ) = #{1 ≤ n ≤ x : for each irreducible factor g of f ,
there exists a prime p > y such that p | g(n)}.
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Inclusion-exclusion on irreducible factors
Friable values of
polynomials
Greg Martin
Proposition
Introduction
Let F be a primitive polynomial, and let F1 , . . . , FK denote
the distinct irreducible factors of F . Then for x ≥ y ≥ 1,
X
X
Ψ(F ; x, y ) = bxc+
(−1)k
M(Fi1 . . . Fik ; x, y ).
1≤k ≤K
1≤i1 <···<ik ≤K
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
If we knew that M(Fi1 . . . Fik ; x, x 1/u ) ∼ x(log du)k , then
X
X
Ψ(F ; x, x 1/u ) ∼ x +
(−1)k
x(log du)k
1≤k ≤K
=x 1+
X
1≤i1 <···<ik ≤K
K
k
(− log du)k
1≤k ≤K
= x(1 − log du)K = xρ(du)K .
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Inclusion-exclusion on irreducible factors
Friable values of
polynomials
Greg Martin
Proposition
Introduction
Let F be a primitive polynomial, and let F1 , . . . , FK denote
the distinct irreducible factors of F . Then for x ≥ y ≥ 1,
X
X
Ψ(F ; x, y ) = bxc+
(−1)k
M(Fi1 . . . Fik ; x, y ).
1≤k ≤K
1≤i1 <···<ik ≤K
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
If we knew that M(Fi1 . . . Fik ; x, x 1/u ) ∼ x(log du)k , then
X
X
Ψ(F ; x, x 1/u ) ∼ x +
(−1)k
x(log du)k
1≤k ≤K
=x 1+
X
1≤i1 <···<ik ≤K
K
k
(− log du)k
1≤k ≤K
= x(1 − log du)K = xρ(du)K .
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Inclusion-exclusion on irreducible factors
Friable values of
polynomials
Greg Martin
Proposition
Introduction
Let F be a primitive polynomial, and let F1 , . . . , FK denote
the distinct irreducible factors of F . Then for x ≥ y ≥ 1,
X
X
Ψ(F ; x, y ) = bxc+
(−1)k
M(Fi1 . . . Fik ; x, y ).
1≤k ≤K
1≤i1 <···<ik ≤K
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
Definition
A uniform version of Hypothesis H
M(f ; x, y ) = #{1 ≤ n ≤ x : for each irreducible factor g of f ,
there exists a prime p > y such that p | g(n)}.
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
1/u
k
We want to prove M(Fi1 . . . Fik ; x, x ) ∼ x(log du) . To do
this, we sort by the values nj = Fij (n)/pj , among those n
counted by M(Fi1 . . . Fik ; x, x 1/u ).
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
DON’T PANIC
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
X
M(f ; x, y ) =
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
not important
Conjecture for friable
values of polynomials
Statement of the conjecture
ξj = fj (x) ≈ x d
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
X
M(f ; x, y ) =
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
not important
Conjecture for friable
values of polynomials
Statement of the conjecture
ηn1 ,...,nk ≈ (y max{n1 , . . . , nk })1/d (n1 · · · nk )−1
Reduction to convenient
polynomials
Translation into prime values of
polynomials
It’s here only because the large primes dividing fj (n) had to
exceed y . (Later we’ll take y = x 1/u .)
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
fairly important
R(f ; n1 , . . . , nk ) = b (mod n1 · · · nk ) :
n1 | f1 (b), n2 | f2 (b), . . . , nk | fk (b)
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
rather important
fn1 ···nk ,b (t) =
Conjecture for friable
values of polynomials
f (n1 · · · nk t + b)
∈ Z[x]
n1 · · · nk
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
In fact, a good understanding of the family fn1 ···nk ,b is
necessary even to treat error terms. However, we’ll only
include the details when treating the main term.
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
rather important
fn1 ···nk ,b (t) =
Conjecture for friable
values of polynomials
f (n1 · · · nk t + b)
∈ Z[x]
n1 · · · nk
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
In fact, a good understanding of the family fn1 ···nk ,b is
necessary even to treat error terms. However, we’ll only
include the details when treating the main term.
Sums of multiplicative functions
Summary
Friable values of
polynomials
Proposition
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
Introduction
M(f ; x, y ) =
X
···
X
X
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
n1 ≤ξ1 /y
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk ) .
n1 · · · nk
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
First: concentrate on
x −b − π(fn1 ···nk ,b ; ηn1 ,...,nk )
π fn1 ···nk ,b ;
n1 · · · nk
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
x −b Look at π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Upper bound sieve (Brun, Selberg):
H(fn1 ···nk ,b )x/n1 · · · nk
x −b
π fn1 ···nk ,b ;
+O
n1 · · · nk
(log x)k +1
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Main term for π(f ; x) (we use Hypothesis UH here!):
H(fn1 ···nk ,b )x
x −b
H(fn1 ···nk ,b ) li fn1 ···nk ,b ;
+O
n1 · · · nk
n1 · · · nk (log x)k +1
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
li is a pretty smooth function:
H(fn1 ···nk ,b )x/n1 · · · nk
H(fn1 ···nk ,b )x
+O
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 · · · nk (log x)k +1
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
x −b Look at π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Upper bound sieve (Brun, Selberg):
H(fn1 ···nk ,b )x/n1 · · · nk
x −b
π fn1 ···nk ,b ;
+O
n1 · · · nk
(log x)k +1
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Main term for π(f ; x) (we use Hypothesis UH here!):
H(fn1 ···nk ,b )x
x −b
H(fn1 ···nk ,b ) li fn1 ···nk ,b ;
+O
n1 · · · nk
n1 · · · nk (log x)k +1
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
li is a pretty smooth function:
H(fn1 ···nk ,b )x/n1 · · · nk
H(fn1 ···nk ,b )x
+O
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 · · · nk (log x)k +1
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
x −b Look at π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Upper bound sieve (Brun, Selberg):
H(fn1 ···nk ,b )x/n1 · · · nk
x −b
π fn1 ···nk ,b ;
+O
n1 · · · nk
(log x)k +1
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Main term for π(f ; x) (we use Hypothesis UH here!):
H(fn1 ···nk ,b )x
x −b
H(fn1 ···nk ,b ) li fn1 ···nk ,b ;
+O
n1 · · · nk
n1 · · · nk (log x)k +1
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
li is a pretty smooth function:
H(fn1 ···nk ,b )x/n1 · · · nk
H(fn1 ···nk ,b )x
+O
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 · · · nk (log x)k +1
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
x −b Look at π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Upper bound sieve (Brun, Selberg):
H(fn1 ···nk ,b )x/n1 · · · nk
x −b
π fn1 ···nk ,b ;
+O
n1 · · · nk
(log x)k +1
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Main term for π(f ; x) (we use Hypothesis UH here!):
H(fn1 ···nk ,b )x
x −b
H(fn1 ···nk ,b ) li fn1 ···nk ,b ;
+O
n1 · · · nk
n1 · · · nk (log x)k +1
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
li is a pretty smooth function:
H(fn1 ···nk ,b )x/n1 · · · nk
H(fn1 ···nk ,b )x
+O
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 · · · nk (log x)k +1
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
Friable values of
polynomials
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
X
X
X
M(f ; x, y ) =
···
n1 ≤ξ1 /y
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
(ni ,nj )=1 (1≤i<j≤k )
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
X
X
X
=
···
H(fn1 ···nk ,b )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
b∈R(f ;n1 ,...,nk )
1 x/n1 · · · nk
×
1+O
.
log(ξ1 /n1 ) · · · log(ξk /nk )
log x
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Now we have:
H(fn1 ···nk ,b )x/n1 · · · nk
H(fn1 ···nk ,b )x
+O
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 · · · nk (log x)k +1
Sums of multiplicative functions
Summary
Understanding M(f ; x, y ) inside out
Friable values of
polynomials
Greg Martin
For f = f1 . . . fk and x and y sufficiently large,
X
X
X
M(f ; x, y ) =
···
n1 ≤ξ1 /y
nk ≤ξk /y b∈R(f ;n1 ,...,nk )
(ni ,nj )=1 (1≤i<j≤k )
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
x −b π fn1 ···nk ,b ;
− π(fn1 ···nk ,b ; ηn1 ,...,nk )
n1 · · · nk
X
X
X
=
···
H(fn1 ···nk ,b )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Introduction
b∈R(f ;n1 ,...,nk )
1 x/n1 · · · nk
×
1+O
.
log(ξ1 /n1 ) · · · log(ξk /nk )
log x
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Next: concentrate on
X
b∈R(f ;n1 ,...,nk )
H(fn1 ···nk ,b )
Summary
Friable values of
polynomials
Nice sums over local solutions
Greg Martin
Recall
Introduction
H(f ) =
Y
p
1
1−
p
−k σ(f ; p)
1−
p
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Recall
Can we have lots of friable values?
Conjecture for prime
values of polynomials
σ(f ; p) = {a (mod p) : f (a) ≡ 0 (mod p)}
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Proposition
Statement of the conjecture
X
Reduction to convenient
polynomials
H(fn1 ···nk ,b ) = H(f )g1 (n1 ) · · · gk (nk ), where
Translation into prime values of
polynomials
b∈R(f ;n1 ,...,nk )
Shepherding the local factors
Sums of multiplicative functions
gj (nj ) =
Y pν knj
σ(f ; p)
1−
p
−1 σ(fj ; pν+1 )
σ(fj ; p ) −
.
p
ν
Summary
Friable values of
polynomials
Nice sums over local solutions
Greg Martin
Recall
Introduction
H(f ) =
Y
p
1
1−
p
−k σ(f ; p)
1−
p
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Recall
Can we have lots of friable values?
Conjecture for prime
values of polynomials
σ(f ; p) = {a (mod p) : f (a) ≡ 0 (mod p)}
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Proposition
Statement of the conjecture
X
Reduction to convenient
polynomials
H(fn1 ···nk ,b ) = H(f )g1 (n1 ) · · · gk (nk ), where
Translation into prime values of
polynomials
b∈R(f ;n1 ,...,nk )
Shepherding the local factors
Sums of multiplicative functions
gj (nj ) =
Y pν knj
σ(f ; p)
1−
p
−1 σ(fj ; pν+1 )
σ(fj ; p ) −
.
p
ν
Summary
Friable values of
polynomials
Nice sums over local solutions
Greg Martin
Recall
Introduction
H(f ) =
Y
p
1
1−
p
−k σ(f ; p)
1−
p
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Recall
Can we have lots of friable values?
R(f ; n1 , . . . , nk ) = b (mod n1 · · · nk ) :
n1 | f1 (b), n2 | f2 (b), . . . , nk | fk (b)
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Proposition
Statement of the conjecture
X
Reduction to convenient
polynomials
H(fn1 ···nk ,b ) = H(f )g1 (n1 ) · · · gk (nk ), where
Translation into prime values of
polynomials
b∈R(f ;n1 ,...,nk )
Shepherding the local factors
Sums of multiplicative functions
gj (nj ) =
Y pν knj
σ(f ; p)
1−
p
−1 σ(fj ; pν+1 )
σ(fj ; p ) −
.
p
ν
Summary
Friable values of
polynomials
Nice sums over local solutions
Greg Martin
Recall
Introduction
H(f ) =
Y
p
1
1−
p
−k σ(f ; p)
1−
p
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Recall
Can we have lots of friable values?
R(f ; n1 , . . . , nk ) = b (mod n1 · · · nk ) :
n1 | f1 (b), n2 | f2 (b), . . . , nk | fk (b)
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Proposition
Statement of the conjecture
X
Reduction to convenient
polynomials
H(fn1 ···nk ,b ) = H(f )g1 (n1 ) · · · gk (nk ), where
Translation into prime values of
polynomials
b∈R(f ;n1 ,...,nk )
Shepherding the local factors
Sums of multiplicative functions
gj (nj ) =
Y pν knj
σ(f ; p)
1−
p
−1 σ(fj ; pν+1 )
σ(fj ; p ) −
.
p
ν
Summary
Friable values of
polynomials
Nice sums over local solutions
Greg Martin
Recall
Introduction
H(f ) =
Y
p
1
1−
p
−k σ(f ; p)
1−
p
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Proving this proposition . . .
Can we have lots of friable values?
. . . is fun, actually, involving the Chinese remainder theorem,
counting lifts of local solutions (Hensel’s lemma), and so on.
Conjecture for prime
values of polynomials
Proposition
Conjecture for friable
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Statement of the conjecture
X
Reduction to convenient
polynomials
H(fn1 ···nk ,b ) = H(f )g1 (n1 ) · · · gk (nk ), where
Translation into prime values of
polynomials
b∈R(f ;n1 ,...,nk )
Shepherding the local factors
Sums of multiplicative functions
gj (nj ) =
Y pν knj
σ(f ; p)
1−
p
−1 σ(fj ; pν+1 )
σ(fj ; pν ) −
.
p
Summary
For f = f1 . . . fk and x and y sufficiently large,
X
X X
M(f ; x, y ) =
···
H(fn1 ···nk ,b )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
b∈R(f ;n1 ,...,nk )
1 x/n1 · · · nk
1+O
log(ξ1 /n1 ) · · · log(ξk /nk )
log x
1 = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
.
log(ξ1 /n1 ) · · · log(ξk /nk )
×
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Therefore: consider
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk
(take care of logarithms later, via
partial summation)
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For f = f1 . . . fk and x and y sufficiently large,
X
X X
M(f ; x, y ) =
···
H(fn1 ···nk ,b )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
b∈R(f ;n1 ,...,nk )
1 x/n1 · · · nk
1+O
log(ξ1 /n1 ) · · · log(ξk /nk )
log x
1 = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
.
log(ξ1 /n1 ) · · · log(ξk /nk )
×
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Therefore: consider
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk
(take care of logarithms later, via
partial summation)
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
For f = f1 . . . fk and x and y sufficiently large,
X
X X
M(f ; x, y ) =
···
H(fn1 ···nk ,b )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
b∈R(f ;n1 ,...,nk )
1 x/n1 · · · nk
1+O
log(ξ1 /n1 ) · · · log(ξk /nk )
log x
1 = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
.
log(ξ1 /n1 ) · · · log(ξk /nk )
×
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Therefore: consider
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk
First: consider more general sums of
multiplicative functions
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Multiplicative functions: one-variable sums
Friable values of
polynomials
Greg Martin
Definition
Let’s say a multiplicative function g(n) is α on average if it
takes nonnegative values and
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
X g(p) log p
∼ α log w.
p
How friable can values of special
polynomials be?
p≤w
How friable can values of general
polynomials be?
Can we have lots of friable values?
ν
Note: we really need upper bounds on g(p ) as well . . .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Lemma
If the multiplicative function g(n) is α on average, then
X g(n)
n≤t
where c(g) =
Y
p
1−
n
1
p
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
∼ c(g)(log t)α ,
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
α 1+
g(p) g(p2 )
+
+
·
·
·
.
p
p2
Multiplicative functions: one-variable sums
Friable values of
polynomials
Greg Martin
Definition
Let’s say a multiplicative function g(n) is α on average if it
takes nonnegative values and
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
X g(p) log p
∼ α log w.
p
How friable can values of special
polynomials be?
p≤w
How friable can values of general
polynomials be?
Can we have lots of friable values?
ν
Note: we really need upper bounds on g(p ) as well . . .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Lemma
If the multiplicative function g(n) is α on average, then
X g(n)
n≤t
where c(g) =
Y
p
1−
n
1
p
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
∼ c(g)(log t)α ,
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
α 1+
g(p) g(p2 )
+
+
·
·
·
.
p
p2
Multiplicative functions: one-variable sums
Friable values of
polynomials
Greg Martin
Definition
Let’s say a multiplicative function g(n) is α on average if it
takes nonnegative values and
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
X g(p) log p
∼ α log w.
p
How friable can values of special
polynomials be?
p≤w
How friable can values of general
polynomials be?
Can we have lots of friable values?
ν
Note: we really need upper bounds on g(p ) as well . . .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Lemma
If the multiplicative function g(n) is α on average, then
X g(n)
n≤t
where c(g) =
Y
p
1−
n
1
p
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
∼ c(g)(log t)α ,
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
α 1+
g(p) g(p2 )
+
+
·
·
·
.
p
p2
Multiplicative functions: one-variable sums
Friable values of
polynomials
Greg Martin
Definition
Let’s say a multiplicative function g(n) is α on average if it
takes nonnegative values and
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
X g(p) log p
∼ α log w.
p
How friable can values of special
polynomials be?
p≤w
How friable can values of general
polynomials be?
Can we have lots of friable values?
ν
Note: we really need upper bounds on g(p ) as well . . .
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Lemma
If the multiplicative function g(n) is α on average, then
X g(n)
n≤t
where c(g) =
Y
p
1−
n
1
p
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
∼ c(g)(log t)α ,
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
α 1+
g(p) g(p2 )
+
+
·
·
·
.
p
p2
Multiplicative functions: more variables
Greg Martin
From previous slide
c(g) =
Y
p
1
1−
p
Friable values of
polynomials
α g(p) g(p2 )
1+
+
+ ···
p
p2
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
By the lemma on the previous slide, we easily get:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Proposition
Can we have lots of friable values?
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
X
n1 ≤t
···
X
nk ≤t
g1 (n1 ) · · · gk (nk )
∼ c(g1 ) · · · c(gk )(log t)k .
n1 · · · nk
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
However, we need the analogous sum with the coprimality
condition (ni , nj ) = 1. (This is where K > 1 makes life
harder!)
Shepherding the local factors
Sums of multiplicative functions
Summary
Multiplicative functions: more variables
Greg Martin
From previous slide
c(g) =
Y
p
1
1−
p
Friable values of
polynomials
α g(p) g(p2 )
1+
+
+ ···
p
p2
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
By the lemma on the previous slide, we easily get:
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Proposition
Can we have lots of friable values?
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
X
n1 ≤t
···
X
nk ≤t
g1 (n1 ) · · · gk (nk )
∼ c(g1 ) · · · c(gk )(log t)k .
n1 · · · nk
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
However, we need the analogous sum with the coprimality
condition (ni , nj ) = 1. (This is where K > 1 makes life
harder!)
Shepherding the local factors
Sums of multiplicative functions
Summary
Multiplicative functions: more variables
Greg Martin
From previous slide
c(g) =
Y
p
1
1−
p
Friable values of
polynomials
α g(p) g(p2 )
1+
+
+ ···
p
p2
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
We get:
How friable can values of general
polynomials be?
Proposition
Can we have lots of friable values?
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
X
···
X
n1 ≤t
nk ≤t
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
∼ c(g1 + · · · + gk )(log t)k .
n1 · · · nk
However, we need the analogous sum with the coprimality
condition (ni , nj ) = 1. (This is where K > 1 makes life
harder!)
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Multiplicative functions: more variables
Greg Martin
From previous slide
c(g) =
Y
p
1
1−
p
Friable values of
polynomials
α g(p) g(p2 )
1+
+
+ ···
p
p2
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
We get:
How friable can values of general
polynomials be?
Proposition
Can we have lots of friable values?
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
X
···
X
n1 ≤t
nk ≤t
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
∼ c(g1 + · · · + gk )(log t)k .
n1 · · · nk
However, we need the analogous sum with the coprimality
condition (ni , nj ) = 1. (This is where K > 1 makes life
harder!)
Never mind that g1 + · · · + gk isn’t multiplicative!
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Partial summation: return of the logs
Greg Martin
The proposition on the previous slide:
. . . gives, after a k -fold partial summation argument:
Introduction
Friable integers
Friable values of polynomials
Proposition
Bounds for friable
values of polynomials
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
∼ c(g1 + · · · + gk )
k
Y
log
j=1
σ(f ;p) −1
σ(f ;p
ξj
.
y
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
2
)
For our functions, gj (p) = 1 − p
σ(fj ; p) − jp
= σ(fj ; p) 1 + O p1 , and σ(fj ; p) is indeed 1 on average by
the prime ideal theorem.
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Partial summation: return of the logs
Greg Martin
The proposition on the previous slide . . .
. . . gives, after a k -fold partial summation argument:
Introduction
Friable integers
Friable values of polynomials
Proposition
Bounds for friable
values of polynomials
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk log(ξ1 /n1 ) · · · log(ξk /nk )
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
∼ c(g1 + · · · + gk )
k
Y
j=1
σ(f ;p) −1
log
log ξj
.
log y
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
σ(f ;p
2
)
For our functions, gj (p) = 1 − p
σ(fj ; p) − jp
= σ(fj ; p) 1 + O p1 , and σ(fj ; p) is indeed 1 on average by
the prime ideal theorem.
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
Partial summation: return of the logs
Greg Martin
The proposition on the previous slide . . .
. . . gives, after a k -fold partial summation argument:
Introduction
Friable integers
Friable values of polynomials
Proposition
Bounds for friable
values of polynomials
If the multiplicative functions g1 (n), . . . , gk (n) are each 1 on
average, then
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
X
···
X
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
g1 (n1 ) · · · gk (nk )
n1 · · · nk log(ξ1 /n1 ) · · · log(ξk /nk )
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
∼ c(g1 + · · · + gk )
k
Y
j=1
σ(f ;p) −1
log
log ξj
.
log y
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
σ(f ;p
2
)
For our functions, gj (p) = 1 − p
σ(fj ; p) − jp
= σ(fj ; p) 1 + O p1 , and σ(fj ; p) is indeed 1 on average by
the prime ideal theorem.
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
For f = f1 . . . fk and x and y sufficiently large,
1 M(f ; x, y ) = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
= H(f )c(g1 + · · · + gk )
Y
k
1 log ξj
log
1+O
.
× x
log y
log x
j=1
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
d
Recall: ξj = fj (x) ≈ x , and we care about y = x
log(log ξj / log y ) ∼ log du.
k
1/u
. Then
We have the order of magnitude x(log du) we wanted . . .
but what about the local factors H(f )c(g1 + · · · + gk )?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
For f = f1 . . . fk and x and y sufficiently large,
1 M(f ; x, y ) = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
= H(f )c(g1 + · · · + gk )
1 (log du)k
1+O
.
× x
log x
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
d
Recall: ξj = fj (x) ≈ x , and we care about y = x
log(log ξj / log y ) ∼ log du.
k
1/u
. Then
We have the order of magnitude x(log du) we wanted . . .
but what about the local factors H(f )c(g1 + · · · + gk )?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
For f = f1 . . . fk and x and y sufficiently large,
1 M(f ; x, y ) = xH(f ) 1 + O
log x
X
X g1 (n1 ) · · · gk (nk )/n1 · · · nk
×
···
log(ξ1 /n1 ) · · · log(ξk /nk )
n1 ≤ξ1 /y
nk ≤ξk /y
(ni ,nj )=1 (1≤i<j≤k )
Greg Martin
Introduction
Friable integers
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
= H(f )c(g1 + · · · + gk )
1 (log du)k
1+O
.
× x
log x
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
d
Recall: ξj = fj (x) ≈ x , and we care about y = x
log(log ξj / log y ) ∼ log du.
k
1/u
. Then
We have the order of magnitude x(log du) we wanted . . .
but what about the local factors H(f )c(g1 + · · · + gk )?
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
The magic moment for H(f )c(g1 + · · · + gk )
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
σ(f ; p)
p
(g1 + · · · + gk )(pν )
and so
pν
We have gj (pν ) =
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
1−
−1 σ(fj ; pν ) −
σ(fj ; p
p
ν+1
)
Can we have lots of friable values?
,
−1 k σ(fj ; pν ) σ(fj ; pν+1 )
1 X
σ(f ; p)
1
−
−
pν
p
pν
pν+1
j=1
−1 σ(f ; p)
σ(f ; pν ) σ(f ; pν+1 )
= 1−
−
p
pν
pν+1
=
since the fj have no common roots modulo p.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
The magic moment for H(f )c(g1 + · · · + gk )
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
σ(f ; p)
p
(g1 + · · · + gk )(pν )
and so
pν
We have gj (pν ) =
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
1−
−1 σ(fj ; pν ) −
σ(fj ; p
p
ν+1
)
Can we have lots of friable values?
,
−1 k σ(fj ; pν ) σ(fj ; pν+1 )
1 X
σ(f ; p)
1
−
−
pν
p
pν
pν+1
j=1
−1 σ(f ; p)
σ(f ; pν ) σ(f ; pν+1 )
= 1−
−
p
pν
pν+1
=
since the fj have no common roots modulo p.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
The magic moment for H(f )c(g1 + · · · + gk )
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
σ(f ; p)
p
(g1 + · · · + gk )(pν )
and so
pν
We have gj (pν ) =
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
1−
−1 σ(fj ; pν ) −
σ(fj ; p
p
ν+1
)
Can we have lots of friable values?
,
−1 k σ(fj ; pν ) σ(fj ; pν+1 )
1 X
σ(f ; p)
1
−
−
pν
p
pν
pν+1
j=1
−1 σ(f ; p)
σ(f ; pν ) σ(f ; pν+1 )
= 1−
−
p
pν
pν+1
=
since the fj have no common roots modulo p.
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
Friable values of
polynomials
The magic moment for H(f )c(g1 + · · · + gk )
Greg Martin
H(f ) =
Y
p
c(g) =
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
Y
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
∞ X
ν=1
A uniform version of Hypothesis H
pν
ν=1
=1+
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
1−
σ(f ; p)
p
−1 Conjecture for friable
values of polynomials
σ(f ; pν ) σ(f ; pν+1 )
−
pν
pν+1
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
ν=1
pν
−1 X
∞ σ(f ; p)
σ(f ; pν ) σ(f ; pν+1 )
=1+ 1−
−
p
pν
pν+1
ν=1
This is a telescoping sum . . .
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
ν=1
pν
−1 X
∞ σ(f ; p)
σ(f ; pν ) σ(f ; pν+1 )
=1+ 1−
−
p
pν
pν+1
ν=1
This is a telescoping sum . . .
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
ν=1
pν
−1 σ(f ; p)
σ(f ; p)
=1+ 1−
p
p
This is a telescoping sum . . . tada!
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
ν=1
pν
−1 σ(f ; p)
σ(f ; p)
=1+ 1−
=
p
p
And this whole expression simplifies . . .
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Therefore
1+
Conjecture for prime
values of polynomials
∞
X
(g1 + · · · + gk )(pν )
ν=1
pν
−1 −1
σ(f ; p)
σ(f ; p)
σ(f ; p)
=1+ 1−
= 1−
.
p
p
p
And this whole expression simplifies . . . nicely.
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
We conclude that
H(f )c(g1 + · · · + gk )
k ∞
Y
X
(g1 + · · · + gk )(pν )
1
= H(f )
1−
1+
p
pν
p
ν=1
k −1
Y
σ(f ; p)
1
1−
= H(f )
1−
p
p
p
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
The magic moment for H(f )c(g1 + · · · + gk )
Friable values of
polynomials
Greg Martin
H(f ) =
Y
p
c(g) =
Y
1
1−
p
−k σ(f ; p)
1−
p
Introduction
Friable integers
Friable values of polynomials
1−
p
1
p
α 1+
g(p) g(p2 )
+
+ ···
p
p2
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
We conclude that
H(f )c(g1 + · · · + gk )
k ∞
Y
X
(g1 + · · · + gk )(pν )
1
= H(f )
1−
1+
p
pν
p
ν=1
k −1
Y
σ(f ; p)
1
=1
1−
= H(f )
1−
p
p
p
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary
. . . amazing!
Summary
Friable values of
polynomials
Greg Martin
Introduction
Friable integers
There are lots of open problems concerning friable
values of polynomials—and many possible
improvements from a single clever new idea.
The asymptotics for friable values of polynomials
depends on the degrees of their irreducible
factors—but shouldn’t depend on the polynomial
otherwise.
Notes to be placed on web page
Friable values of polynomials
Bounds for friable
values of polynomials
How friable can values of special
polynomials be?
How friable can values of general
polynomials be?
Can we have lots of friable values?
Conjecture for prime
values of polynomials
Schinzel’s “Hypothesis H”
(Bateman–Horn conjecture)
A uniform version of Hypothesis H
Conjecture for friable
values of polynomials
Statement of the conjecture
Reduction to convenient
polynomials
www.math.ubc.ca/∼gerg/talks.html
Translation into prime values of
polynomials
Shepherding the local factors
Sums of multiplicative functions
Summary