Affiliation:
1. Department of Mathematics , University at Buffalo , 244 Mathematics Building Buffalo , NY 14260-2900 , USA
Abstract
Abstract
The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for
X
2
3
+
ϵ
<
H
<
X
1
-
ϵ
{X^{\frac{2}{3}+\epsilon}<H<X^{1-\epsilon}}
there are constants
B
h
{B_{h}}
such that
∑
X
≤
n
≤
2
X
λ
f
(
n
)
2
λ
f
(
n
+
h
)
2
-
B
h
X
=
O
f
,
A
,
ϵ
(
X
(
log
X
)
-
A
)
\sum_{X\leq n\leq 2X}\lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,%
\epsilon}(X(\log X)^{-A})
for all but
O
f
,
A
,
ϵ
(
H
(
log
X
)
-
3
A
)
{O_{f,A,\epsilon}(H(\log X)^{-3A})}
integers
h
∈
[
1
,
H
]
{h\in[1,H]}
where
{
λ
f
(
n
)
}
n
≥
1
{\{\lambda_{f}(n)\}_{n\geq 1}}
are normalized Hecke eigenvalues of a fixed holomorphic cusp form f.
Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts
m
1
{m_{1}}
and
m
2
{m_{2}}
. In order to treat
m
2
{m_{2}}
, we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat
m
1
{m_{1}}
, we apply Parseval’s identity and Gallagher’s lemma.
Subject
Applied Mathematics,General Mathematics
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Cited by
2 articles.
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