Shifted convolution sums for $${{\varvec{SL}}}(m)$$ SL ( m )
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Guangwei Hu
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
· Guangshi Lü
Shifted convolution sums for SL(m) Received: 18 March 2019 / Accepted: 8 November 2019 Abstract. In this paper, we study the shifted convolution sums of the Fourier coefficients λπ (1, . . . , 1, n) and rs,k (n) with k ≥ 3, where rs,k (n) denotes the number of representations of the positive integer n as sums of s kth powers. We are able to generalize or improve previous results.
1. Introduction Suppose a1 (n) and a2 (n) are two arithmetic functions, and let b ≥ 0 be an integer. It is a classical and important problem in analytic number theory to study the shifted convolution sum a1 (n)a2 (n + b). n≤x
For the divisor function d(n), Ingham [3] established the asymptotic formula x
d(n)d(n + k) ∼
n=1
6 σ−1 (k)x log2 x, π2
where σ−1 (n) = d|n d1 . In 1995, Pitt [12] firstly succeeded in giving non-trivial bounds for a G L(3) × G L(2) shifted convolution sum, which states that for positive integer r , 71 d3 (n)λ f (r n − 1) ≤ x 72 +ε n≤x
x 1/24 , where d3 (n) denotes the number of representations of
uniformly in 0 < r < n as the product of three natural numbers, and λ f (n) are the normalized coefficients of holomorphic cusp forms for S L(2, Z). Recently, Munshi [11] improved Pitt’s result to 2 34 d3 (n)λ f (r n − 1) ≤ r 7 x 35 +ε n≤x
G. Hu (B) · G. Lü: School of Mathematics, Shandong University, Jinan 250100, China. e-mail: [email protected] G. Lü e-mail: [email protected] Mathematics Subject Classification: 11E76 · 11F30 · 11P55
https://doi.org/10.1007/s00229-019-01166-1
G. Hu, G. Lü
uniformly in 0 < r < x 1/10 . Let λπ (1, . . . , 1, n) be the normalized Fourier coefficients of an even Hecke– Maass form π for S L(m, Z). Due to the work of Kim and Sarnak [6] (m ≤ 4) and Luo, Rudnick and Sarnak [9] (m ≥ 5), one has |λπ (1, . . . , 1, n)| ≤ n θm dm (n),
(1.1)
where dm (n) denotes the number of representations of n as the product of m natural numbers, and θ3 =
5 9 1 1 , θ4 = , θm = − 2 (m ≥ 5). 14 22 2 m +1
The generalized Ramanujan conjecture asserts that θm = 0. Let rs,k (n) denote the number of representations of the positive integer n as sums of s kth powers, namely, rs,k (n) = # (m 1 , m 2 , . . . , m s ) ∈ Zs : m k1 + m k2 + · · · + m ks = n . For k = 2, Luo [8] derived a sharp bound for the shifted convolution sum convolving the Fourier coefficients of holomorphic cusp forms with the Fourier coefficients rs,2 (n) of theta series. Recently, Lü et al. [7] introduced some simple approaches to improve Luo’s result, and showed that s λ f (n + b)rs,2 (n) f,s,ε x 2 −ϑs +ε n≥1
uniformly for x ≥ 2 and 0 ≤ b ≤ x, where ϑ3 = 1/4 and ϑs = 1/2 (s ≥ 4). As a generalization, Jiang and Lü [5] showed a similar result for the Fourier coefficients λπ (1, . . . , 1, n) of a G L(m) Hecke–Maass form, namely,
2−θm
3
λπ (1, . . . , 1, n + b)r3,2 (n) π,ε x 2 − 2(m+4+2θm )+1 +ε .
n≥1
Further, Jiang and Lü [5] also considered an analogous shifted convolution sums in a more general case. And they obtai
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