On the random Chowla conjecture
We show that for a Steinhaus random multiplicative function f : N → D and any polynomial P ( x ) ∈ Z [ x ] of deg P ≥ 2 which is not of the form w ( x + c ) d for some w ∈ Z , c ∈ Q , we have 1 N ∑ n ≤ N f ( P ( n ) ) → d C N ( 0 , 1 ) , where C N ( 0 , 1 ) is the standard complex Gaussian distribut...
Saved in:
| Published in | Geometric and functional analysis Vol. 33; no. 3; pp. 749 - 777 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.06.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Summary: | We show that for a Steinhaus random multiplicative function
f
:
N
→
D
and any polynomial
P
(
x
)
∈
Z
[
x
]
of
deg
P
≥
2
which is not of the form
w
(
x
+
c
)
d
for some
w
∈
Z
,
c
∈
Q
, we have
1
N
∑
n
≤
N
f
(
P
(
n
)
)
→
d
C
N
(
0
,
1
)
,
where
C
N
(
0
,
1
)
is the standard complex Gaussian distribution with mean 0 and variance 1. This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of
x
≥
1
, such that
∑
n
≤
x
f
(
P
(
n
)
)
≫
P
x
(
log
log
x
)
1
/
2
,
for any polynomial
P
(
x
)
∈
Z
[
x
]
with
deg
P
≥
2
,
which is not a product of linear factors (over
Q
). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear polynomial
P
(
n
)
=
n
,
where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be
O
(
x
(
log
log
x
)
1
4
+
ε
)
for any
ε
>
0
. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1016-443X 1420-8970 |
| DOI: | 10.1007/s00039-023-00641-y |