The least common multiple
of sequences of integers
Ana Zumalacárregui
H
Carl Pomerance’s 70’s
June 2015, Athens, GA
Joint work with J. Cilleruelo, J. Rué and P. Šarka.
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m)
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
Theorem (Chebyshev, 1850)
π(n) ∼
n
⇔ ψ(n) ∼ n.
log n
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
Theorem (Chebyshev, 1850)
π(n) ∼
n
⇔ ψ(n) ∼ n.
log n
Prime Number Theorem (Hadamard,
de la Vallée-Poussin, 1896)
ψ(n) ∼ n.
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
Theorem (Chebyshev, 1850)
π(n) ∼
n
⇔ ψ(n) ∼ n.
log n
Prime Number Theorem (Hadamard,
de la Vallée-Poussin, 1896)
What if one considers a different sequence?
ψ(n) ∼ n.
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
Theorem (Chebyshev, 1850)
π(n) ∼
n
⇔ ψ(n) ∼ n.
log n
Prime Number Theorem (Hadamard,
de la Vallée-Poussin, 1896)
ψ(n) ∼ n.
What if one considers a different sequence?
. Arithmetic progressions: if (a, b) = 1, PNT for AP’s
X 1
a
log lcm(a + b, . . . an + b) ∼ n
.
(Bateman’02)
ϕ(a)
r
(r ,a)=1
The prime number theorem
On his way to prove the PNT, Chebyshev introduced the function:
P
ψ(n) = m≤n Λ(m) = log lcm(1, 2, . . . , n)
Theorem (Chebyshev, 1850)
π(n) ∼
n
⇔ ψ(n) ∼ n.
log n
Prime Number Theorem (Hadamard,
de la Vallée-Poussin, 1896)
ψ(n) ∼ n.
What if one considers a different sequence?
. Arithmetic progressions: if (a, b) = 1, PNT for AP’s
X 1
a
log lcm(a + b, . . . an + b) ∼ n
.
(Bateman’02)
ϕ(a)
r
(r ,a)=1
. Quadratic sequences:
Theorem (Cilleruelo’11) if f ∈ Z[x] quadratic, reducible
log lcm(f (1), . . . , f (n)) ∼ Cf n.
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Cilleruelo’11) If f ∈ Z[x] is quadratic and irreducible, then
log lcm (f (1), . . . , f (n)) = n log n + Bf n + o(n).
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Cilleruelo’11) If f (x) = x 2 + 1 then
log lcm (12 + 1, . . . , n2 + 1) = n log n + Bn + o(n),
Where
B =γ−1−
log 2
2
−
P
p6=2
(−1)
p−1
2
log p
p−1
= −0.066275634213060706383563177025...
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Cilleruelo’11) If f (x) = x 2 + 1 then
log lcm (12 + 1, . . . , n2 + 1) = n log n + Bn + o(n),
Where
B =γ−1−
log 2
2
−
P
p6=2
(−1)
p−1
2
log p
p−1
= −0.066275634213060706383563177025...
Sketch of the proof:
Qn
. Approximate Ln = lcm(12 + 1, . . . , n2 + 1) by Pn = k=1 (k 2 + 1).
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Cilleruelo’11) If f (x) = x 2 + 1 then
log lcm (12 + 1, . . . , n2 + 1) = n log n + Bn + o(n),
Where
B =γ−1−
log 2
2
−
P
p6=2
(−1)
p−1
2
log p
p−1
= −0.066275634213060706383563177025...
Sketch of the proof:
Qn
. Approximate Ln = lcm(12 + 1, . . . , n2 + 1) by Pn = k=1 (k 2 + 1).
. Study the excess on every prime p|Pn :
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Cilleruelo’11) If f (x) = x 2 + 1 then
log lcm (12 + 1, . . . , n2 + 1) = n log n + Bn + o(n),
Where
B =γ−1−
log 2
2
−
P
p6=2
(−1)
p−1
2
log p
p−1
= −0.066275634213060706383563177025...
Sketch of the proof:
Qn
. Approximate Ln = lcm(12 + 1, . . . , n2 + 1) by Pn = k=1 (k 2 + 1).
. Study the excess on every prime p|Pn :
Theorem (Duke-Friedlander-Iwaniec’95, Tóth’00) Let f ∈ Z[x] be an
irreducible quadratic polynomial. The sequence
{{ν/p} : 0 ≤ ν < p, f (ν) ≡ 0 (mod p)}
is uniformly distributed when p runs over the set of prime numbers.
The lcm of an irreducible quadratic sequence
Obs: No analogue of the PNT for primes in quadratic sequences.
Theorem (Rué-Šarka-Z.) For every θ < 4/9 we have log lcm (12 + 1, . . . , n2 + 1) = n log n + Bn + O lognθ n .
I Thanks to the estimate of the discrepancy for { νp } due to Homma’08.
Sketch of the proof:
Qn
. Approximate Ln = lcm(12 + 1, . . . , n2 + 1) by Pn = k=1 (k 2 + 1).
. Study the excess on every prime p|Pn :
Theorem (Duke-Friedlander-Iwaniec’95, Tóth’00) Let f ∈ Z[x] be an
irreducible quadratic polynomial. The sequence
{{ν/p} : 0 ≤ ν < p, f (ν) ≡ 0 (mod p)}
is uniformly distributed when p runs over the set of prime numbers.
What about cubic sequences?
Nothing is known for irreducible polynomials of higher degree:
What about cubic sequences?
Nothing is known for irreducible polynomials of higher degree:
Conjecture (Cilleruelo) Let f ∈ Z[x] be an irreducible polynomial of
degree d ≥ 2. Then
log lcm (f (1), . . . , f (n)) ∼ (d − 1)n log n.
a
The lcm of a random set
For a subset S ⊂ {1, . . . , n} chosen uniformly at random amongst all the
possible 2n sets,
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
The lcm of a random set
For a subset S ⊂ {1, . . . , n} chosen uniformly at random amongst all the
possible 2n sets,
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.)
For almost every subset S ⊆ [n]: log lcm(s : s ∈ S) = n log 2 + o(n).
In other words,
lcm (s : s ∈ S) = 2n(1+o(1))
for almost every S ⊂ {1, . . . , n}.
The lcm of a random set
For a subset S ⊂ {1, . . . , n} chosen uniformly at random amongst all the
possible 2n sets,
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.)
For almost every subset S ⊆ [n]: log lcm(s : s ∈ S) = n log 2 + o(n).
In other words,
lcm (s : s ∈ S) = 2n(1+o(1))
for almost every S ⊂ {1, . . . , n}.
In general, we study the random variable log lcm(s : s ∈ S) where:
every element s ∈ S is independently chosen with fixed probability δ ∈ (0, 1).
The lcm of a random set
For a subset S ⊂ {1, . . . , n} chosen uniformly at random amongst all the
possible 2n sets,
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.)
For almost every subset S ⊆ [n]: log lcm(s : s ∈ S) = n log 2 + o(n).
In other words,
lcm (s : s ∈ S) = 2n(1+o(1))
for almost every S ⊂ {1, . . . , n}.
In general, we study the random variable log lcm(s : s ∈ S) where:
every element s ∈ S is independently chosen with fixed probability δ ∈ (0, 1).
Theorem (Cilleruelo-Rué-Šarka-Z.) If δ = δ(n) < 1, and δn → ∞,
log lcm(s : s ∈ S) ∼
nδ log(δ −1 )
1−δ
a. a. s.
The lcm of a random set of fixed size
Observation:
log lcm(f (k) : f (k) ≤ n) =
n1/2 log(n1/2 ) + Bn1/2 + o(n1/2 ),
n1/2 + o(n1/2 ),
f (x) = x 2 + 1,
f (x) = x 2 − 1.
The lcm of a random set of fixed size
Observation:
log lcm(f (k) : f (k) ≤ n) =
n1/2 log(n1/2 ) + Bn1/2 + o(n1/2 ),
n1/2 + o(n1/2 ),
f (x) = x 2 + 1,
f (x) = x 2 − 1.
If we choose at random a set with ∼ n1/2 elements in {1, . . . , n}
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
The lcm of a random set of fixed size
Observation:
log lcm(f (k) : f (k) ≤ n) =
n1/2 log(n1/2 ) + Bn1/2 + o(n1/2 ),
n1/2 + o(n1/2 ),
f (x) = x 2 + 1,
f (x) = x 2 − 1.
If we choose at random a set with ∼ n1/2 elements in {1, . . . , n}
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.) For almost every set S ⊆ [n] with
n1/2 elements:
log lcm(s : s ∈ S) ∼ n1/2 log(n1/2 ).
The lcm of a random set of fixed size
Observation:
log lcm(f (k) : f (k) ≤ n) =
n1/2 log(n1/2 ) + Bn1/2 + o(n1/2 ),
n1/2 + o(n1/2 ),
f (x) = x 2 + 1,
f (x) = x 2 − 1.
If we choose at random a set with ∼ n1/2 elements in {1, . . . , n}
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.) For almost every set S ⊆ [n] with
n1/2 elements:
log lcm(s : s ∈ S) ∼ n1/2 log(n1/2 ).
We study the random variable log lcm(s : s ∈ S) where:
S ⊂ [n] is chosen uniformly at random amongst all sets with k elements.
Aa
The lcm of a random set of fixed size
Observation:
log lcm(f (k) : f (k) ≤ n) =
n1/2 log(n1/2 ) + Bn1/2 + o(n1/2 ),
n1/2 + o(n1/2 ),
f (x) = x 2 + 1,
f (x) = x 2 − 1.
If we choose at random a set with ∼ n1/2 elements in {1, . . . , n}
AAAAwhat could we say about the quantity log lcm(s : s ∈ S)?
Theorem (Cilleruelo-Rué-Šarka-Z.) For almost every set S ⊆ [n] with
n1/2 elements:
log lcm(s : s ∈ S) ∼ n1/2 log(n1/2 ).
We study the random variable log lcm(s : s ∈ S) where:
S ⊂ [n] is chosen uniformly at random amongst all sets with k elements.
Aa
Theorem (Cilleruelo-Rué-Šarka-Z.) If k = k(n) < n with k → ∞, then
for almost every S ∈ [n]
k :
√
k log(n/k) log lcm(s : s ∈ S) =
1 + O e −C log k .
1 − k/n
What about cubic sequences?
Nothing is known for irreducible polynomials of higher degree:
Conjecture (Cilleruelo) Let f ∈ Z[x] be an irreducible polynomial of
degree d ≥ 2. Then
log lcm (f (1), . . . , f (n)) ∼ (d − 1)n log n.
Corollary (Cilleruelo-Rué-Šarka-Z.) For almost every sequence
S ⊆ {1, . . . , f (n)} with n elements, we have
log lcm(s : s ∈ S) ∼ (d − 1)n log n.
The least common multiple
of sequences of integers
Ana Zumalacárregui
H
Carl Pomerance’s 70’s
June 2015, Athens GA
Joint work with J. Cilleruelo, J. Rué and P. Šarka.