Multiplicative functions and nilsequences
Lilian Matthiesen
Universität Hannover, Germany
Classical setting:
1X
f (n)e(nα),
x n6x
where f : N → C is multiplicative.
Classical setting:
1X
f (n)e(nα),
x n6x
where f : N → C is multiplicative.
1 Motivation and related work
2 Classical results
3 Generalisation to unbounded functions
4 Generalisation to nilsequences
5 Main theorem and applications
Part 1 - Motivation
Green and Tao’s nilpotent Hardy-Littlewood method
Part 1 - Motivation
Green and Tao’s nilpotent Hardy-Littlewood method
Aim: Asymptotic formulae for linear correlations of arithmetic
functions
Part 1 - Motivation
Green and Tao’s nilpotent Hardy-Littlewood method
Aim: Asymptotic formulae for linear correlations of arithmetic
functions
ϕ1 , . . . , ϕr ∈ Z[X1 , . . . , Xs ]
a1 , . . . , ar ∈ Z
h1 , . . . , hr : Z → C
linear forms,
pairwise non-proportional
Part 1 - Motivation
Green and Tao’s nilpotent Hardy-Littlewood method
Aim: Asymptotic formulae for linear correlations of arithmetic
functions
ϕ1 , . . . , ϕr ∈ Z[X1 , . . . , Xs ]
linear forms,
pairwise non-proportional
a1 , . . . , ar ∈ Z
h1 , . . . , hr : Z → C
Consider the linear correlation
X
h1 (ϕ1 (x) + a1 ) . . . hr (ϕr (x) + ar )
x∈Zs ∩tK
where K ⊂ [−1, 1]s , convex.
(as t → ∞)
Part 1 - Motivation
Green and Tao’s nilpotent Hardy-Littlewood method
Aim: Asymptotic formulae for linear correlations of arithmetic
functions
ϕ1 , . . . , ϕr ∈ Z[X1 , . . . , Xs ]
linear forms,
pairwise non-proportional
a1 , . . . , ar ∈ Z
h1 , . . . , hr : Z → C
Consider the linear correlation
X
Y
h1 (ϕ1 (x) + a1 ) . . . hr (ϕr (x) + ar ) = ts β∞
βp + o(ts ),
x∈Zs ∩tK
where K ⊂ [−1, 1]s , convex.
p
(as t → ∞)
Green and Tao’s nilpotent Hardy-Littlewood method
Such asymptotic formulae hold, provided
1. there are simultaneous pseudo-random majorants for the
functions h1 |{1,...,t} , . . . , hr |{1,...,t} , t → ∞;
2. as t → ∞,
hi {1,...,t}
−
1 X
t
hi (m) ,
m6t
is orthogonal to nilsequences.
(1 6 i 6 r),
Polynomial nilsequences
G a connected, simply connected k-step nilpotent Lie Group.
Γ a discrete, co-compact subgroup.
Then G/Γ is a k-step nilmanifold. dG/Γ a smooth metric.
P (n)
g : Z → G is a polynomial sequence if g(n) = a1 1
P1 , . . . , Pq ∈ Z[X] and a1 , . . . , aq ∈ G.
Polynomial nilsequence:
P (n)
. . . aq q
(F(g(n)Γ))n∈Z
where F : G/Γ → C is continuous, Lipschitz and kFk∞ 6 1.
for
Polynomial nilsequences
G a connected, simply connected k-step nilpotent Lie Group.
Γ a discrete, co-compact subgroup.
Then G/Γ is a k-step nilmanifold. dG/Γ a smooth metric.
P (n)
g : Z → G is a polynomial sequence if g(n) = a1 1
P1 , . . . , Pq ∈ Z[X] and a1 , . . . , aq ∈ G.
Polynomial nilsequence:
P (n)
. . . aq q
for
(F(g(n)Γ))n∈Z
where F : G/Γ → C is continuous, Lipschitz and kFk∞ 6 1.
Aim to show that for 1 6 i 6 r and for all poly. nilsequences:
1 X
1 X
h
(n)
−
h
(m)
F(g(n)Γ)
i
i
N
N
n6N
m6N
1 X
= oG/Γ
|hi (m)|
N
m6N
Part 1 - Related work
Aim:
1 X
1 X
hi (m) F(g(n)Γ)
hi (n) −
N
N
n6N
m6N
1 X
= oG/Γ
|hi (m)|
N
m6N
Part 1 - Related work
Aim:
1 X
1 X
hi (m) F(g(n)Γ)
hi (n) −
N
N
n6N
m6N
1 X
= oG/Γ
|hi (m)|
N
m6N
Frantzikinakis–Host [’14]:
Structure theorem for bounded multiplicative functions.
Error term small over space of all multiplicative functions.
Frantzikinakis–Host [’15]:
Asymptotics for linear correlations of bounded hi with the
property that
1 X
hi (n) ∼ ci .
N
n6N
Part 2 - Classical results
Theorem (Daboussi, 1974)
Let f : N → C be multiplicative, kf k∞ 6 1, and let α be irrational.
Then
1 X
f (n)e(αn) → 0
as
N → ∞.
N
n6N
Part 2 - Classical results
Theorem (Daboussi, 1974)
Let f : N → C be multiplicative, kf k∞ 6 1, and let α be irrational.
Then
1 X
f (n)e(αn) → 0
as
N → ∞.
N
n6N
Theorem (Montgomery and Vaughan, 1977)
Let f : N → C be multiplicative and suppose
|f (p)| 6 H
at primes
and
1 X
|f (n)|2 6 H2 .
N
n6N
If |α −
a
q|
<
q−2 ,
where (a, q) = 1 and 2 6 R 6 q 6 N/R,
Part 2 - Classical results
Theorem (Daboussi, 1974)
Let f : N → C be multiplicative, kf k∞ 6 1, and let α be irrational.
Then
1 X
f (n)e(αn) → 0
as
N → ∞.
N
n6N
Theorem (Montgomery and Vaughan, 1977)
Let f : N → C be multiplicative and suppose
|f (p)| 6 H
at primes
and
1 X
|f (n)|2 6 H2 .
N
n6N
If |α −
a
q|
<
q−2 ,
where (a, q) = 1 and 2 6 R 6 q 6 N/R, then
1 X
1
(log R)3/2
√
f (n)e(αn) H
+
.
N
log N
R
n6N
Montgomery and Vaughan, sketch of proof
X
n6N
f (n)e(nα)
Montgomery and Vaughan, sketch of proof
X
n6N
f (n)e(nα) log(N/n)
Montgomery and Vaughan, sketch of proof
X
1/2 X
1/2
X
(log N/n)2
|f (n)|2
f (n)e(nα) log(N/n) 6
n6N
n6N
|
n6N
{z
6N1/2
}
Montgomery and Vaughan, sketch of proof
X
1/2 X
1/2
X
(log N/n)2
|f (n)|2
f (n)e(nα) log(N/n) 6
n6N
n6N
|
n6N
{z
}
6N1/2
1 X
1/2
1 1 X
f (n)e(nα) |f (n)|2
Thus, N
log N N
n6N
n6N
1 1 X
+
f
(n)e(nα)
log
n
log N N
n6N
Montgomery and Vaughan, sketch of proof
X
1/2 X
1/2
X
(log N/n)2
|f (n)|2
f (n)e(nα) log(N/n) 6
n6N
n6N
|
n6N
{z
6N1/2
}
1 X
1/2
1 1 X
f (n)e(nα) |f (n)|2
Thus, N
log N N
n6N
n6N
1 1 X
+
f
(n)e(nα)
log
n
log N N
n6N
P
Recall log n = d|n Λ(d).
Montgomery and Vaughan, sketch of proof
X
1/2 X
1/2
X
(log N/n)2
|f (n)|2
f (n)e(nα) log(N/n) 6
n6N
n6N
|
n6N
{z
6N1/2
}
1 X
1/2
1 1 X
f (n)e(nα) |f (n)|2
Thus, N
log N N
n6N
n6N
1 1 X
+
f
(n)e(nα)
log
n
log N N
n6N
P
Recall log n = d|n Λ(d). Suffices to bound
1 X
f (nm)Λ(m)e(nmα)
N
nm6N
1 X
f (n)f (p)Λ(p)e(pnα)
N
np6N
Montgomery and Vaughan, sketch of proof
1 X
f (n)f (p)Λ(p)e(pnα)
N
np6N
Montgomery and Vaughan, sketch of proof
1 X
f (n)f (p)Λ(p)e(pnα)
N
np6N
6
1 X
|f (n)|2
N
n6N
1 X
N
p,p0
1/2
×
f (p)f (p0 ) log(p) log(p0 )
X
n
e((p − p0 )nα)
1/2
Montgomery and Vaughan, sketch of proof
1 X
f (n)f (p)Λ(p)e(pnα)
N
np6N
6
1 X
|f (n)|2
N
n6N
1 X
N
p,p0
1/2
×
f (p)f (p0 ) log(p) log(p0 )
X
e((p − p0 )nα)
1/2
n
use standard sieve estimate to bound #{(p, p0 ) : p − p0 = h}
+ standard exponential sum estimate
+ delicate decomposition of summation ranges for n, p, p0
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
|f (n)|2 6 H2 .
N
n6N
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
2
2
|f (n)| 6 H .
N
n6N
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
2
2
|f (n)| 6 H .
N
n6N
f = f1 ∗ · · · ∗ fH
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
2
2
|f (n)| 6 H .
N
n6N
f = f1 ∗ · · · ∗ fH
where
for 1 6 i 6 H − 1
(
f (p)/H
fi (pk ) =
0
if k = 1
if k > 1
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
2
2
|f (n)| 6 H .
N
n6N
f = f1 ∗ · · · ∗ fH
where
(
f (p)/H
fi (pk ) =
0
if k = 1
if k > 1
for 1 6 i 6 H − 1 and fH is defined via f = f1 ∗ · · · ∗ fH .
Part 3 - Generalisation to unbounded functions
Suppose
|f (p)| 6 H
at primes,
1 X
2
2
|f (n)| 6 H .
N
n6N
f = f1 ∗ · · · ∗ fH
where
(
f (p)/H
fi (pk ) =
0
if k = 1
if k > 1
for 1 6 i 6 H − 1 and fH is defined via f = f1 ∗ · · · ∗ fH .
Note:
(
16i6H−1
|fi (n)| 6 1 if
.
i = H and n square-free
Part 3 - Generalisation to unbounded functions
Lemma
Let f be a non-negative multiplicative function that satisfies
f (pk ) 6 Hk at all prime powers pk . Suppose
X
f (p) log p x
p6x
and
X f (p) 1X
1
exp
.
f (n) x n6x
log x
p
p6x
Then there is a decomposition f = f1 ∗ · · · ∗ fH such that
s
1X
1X
|fi (n)|2 (log x)1/2
|fi (n)|
x n6x
x n6x
for 1 6 i 6 H.
Part 3 - Generalisation to unbounded functions
How can we employ the Dirichlet decomposition?
X
X
f (n)e(nα) =
f1 (n1 ) . . . fH (nH )e(n1 . . . nH α)
n6N
n1 ...nH 6N
Part 3 - Generalisation to unbounded functions
How can we employ the Dirichlet decomposition?
X
X
f (n)e(nα) =
f1 (n1 ) . . . fH (nH )e(n1 . . . nH α)
n6N
n1 ...nH 6N
Use Hyperbola method
H
X
i=1
X
X
Di 6N1−1/H d1 ···dbi ···dH
=Di
Y
j6=i
fj (dj )
X
ni 6N/Di
fi (ni )e(ni Di α)
Part 4 - From exponentials to nilsequences
Aim: understand
1 X
1 X
f (m) F(g(n)Γ)
f (n) −
N
N
n6N
m6N
for multiplicative f .
Problem: Know much less about F(g(n)Γ) than about e(αn).
Part 4 - From exponentials to nilsequences
Aim: understand
1 X
1 X
f (m) F(g(n)Γ)
f (n) −
N
N
n6N
m6N
for multiplicative f .
Problem: Know much less about F(g(n)Γ) than about e(αn).
[Factorisation theorem and W-trick]
Part 4 - From exponentials to nilsequences
Aim: understand
1 X
1 X
f (m) F(g(n)Γ)
f (n) −
N
N
n6N
m6N
for multiplicative f .
Problem: Know much less about F(g(n)Γ) than about e(αn).
[Factorisation theorem and W-trick]
Recall: g : {1, . . . , N} → G is totally δ-equidistributed in G/Γ
(equipped with Haar measure) if
Z
1 X
F(g(n)Γ) −
F 6 δkFkLip
|P|
G/Γ
n∈P
for all Lipschitz functions F : G/Γ → C and progressions
P ⊂ {1, . . . , N} of length |P| > δN.
Part 4 - From exponentials to nilsequences
X
mp6N
f (m)f (p)Λ(p)F(g(mp)Γ)
Part 4 - From exponentials to nilsequences
X
f (m)f (p)Λ(p)F(g(mp)Γ)
mp6N
X X
=
f (m)f (p)Λ(p)F(g(mp)Γ)
m6X p6N/m
+
X X
m>X p6N/m
f (m)f (p)Λ(p)F(g(mp)Γ)
Part 4 - From exponentials to nilsequences
X
f (m)f (p)Λ(p)F(g(mp)Γ)
mp6N
X X
=
f (m)f (p)Λ(p)F(g(mp)Γ)
m6X p6N/m
+
X X
f (m)f (p)Λ(p)F(g(mp)Γ)
m>X p6N/m
Applying CS to both terms to separate f and F , yields
X
1/2 X
1/2
X
|f (p)|2 Λ(p)
f (m)f (m0 )
Λ(p)F(g(mp)Γ)F(g(m0 p)Γ)
m,m0
p6N
p
and
X
1/2 X
1/2
X
|f (m)|2
f (p)f (p0 )Λ(p)Λ(p0 )
F(g(pm)Γ)F(g(p0 m)Γ)
m6N
p,p0
m
Part 5 - Main result (mod W-trick)
Let MH denote the set of multiplicative functions f : N → R
such that there is a decomposition f = f1 ∗ · · · ∗ ft for t 6 H
with properties:
1 |fi (pk )| 6 H k at all prime powers.
2 there is θf ∈ (0, 1] such that
s
1X
1X
|fi (n)|2 (log x)1−θf
|fi (n)|
x n6x
x n6x
3
if x x0C1 xC2 for positive constants C1 , C2 , then
1 X
1X
fi (n)
fi (n) C1 ,C2 0
x n6x
x
0
n6x
4
1
log x
X
j∈N
exp(log log x)2 <2j 6x
1 X
1X
f
(n)
|fi (n)|
i
x n6x
2j
j
n62
Part 5 - Main result (mod W-trick)
If f = f1 ∗ · · · ∗ ft ∈ MH , if G/Γ is a nilmanifold and if
(F(g(n)Γ))n∈Z is a polynomial nilsequences, then:
1 X
f (n) − Sf (N) F(g(n)Γ)
N
n6N
= oG/Γ (1 + kFkLip )S|f1 |∗···∗|ft | (N),
where
Sf (N) =
1 X
f (n).
N
m6N
Part 5 - Main result (mod W-trick)
If f = f1 ∗ · · · ∗ ft ∈ MH , if G/Γ is a nilmanifold and if
(F(g(n)Γ))n∈Z is a polynomial nilsequences, then:
1 X
f (n) − Sf (N) F(g(n)Γ)
N
n6N
= oG/Γ (1 + kFkLip )S|
where
Sf (N) =
1 X
f (n).
N
m6N
f
| (N),
Part 5 - Main result (mod W-trick)
If f = f1 ∗ · · · ∗ ft ∈ MH , if G/Γ is a nilmanifold and if
(F(g(n)Γ))n∈Z is a polynomial nilsequences, then:
1 X
f (n) − Sf (N) F(g(n)Γ)
N
n6N
= oG/Γ (1 + kFkLip )S| f1 ∗···∗ft | (N),
where
Sf (N) =
1 X
f (n).
N
m6N
Part 5 - Main result (mod W-trick)
If f = f1 ∗ · · · ∗ ft ∈ MH , if G/Γ is a nilmanifold and if
(F(g(n)Γ))n∈Z is a polynomial nilsequences, then:
1 X
f (n) − Sf (N) F(g(n)Γ)
N
n6N
= oG/Γ (1 + kFkLip )S|f1 |∗···∗|ft | (N),
where
Sf (N) =
1 X
f (n).
N
m6N
Part 5 - Applications
This allows us to asymptotically evaluate
X
h1 (ϕ1 (x) + a1 ) . . . hr (ϕr (x) + ar )
x∈Zs ∩tK
for h1 , . . . , hr could, for instance, be
characteristic function of set of sums of two squares,
characteristic function of set of numbers composed of
primes that split completely in a given Galois extension
K/Q of finite degree,
general divisor function dk = |1 ∗ ·{z
· · ∗ 1},
k
positive multiplicative function with f (pk ) 6 Hk ,
f (n) ε nε and µ ∗ f > 0.