Affiliation:
1. School of Mathematics and Statistics, Shandong University , Weihai, Shandong 264209, China
2. School of Mathematics, Shandong University , Jinan, Shandong 250100, China
Abstract
Abstract
Let $\lambda _{\phi }(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke–Maass cusp form on $\textrm{SL}_{2}(\mathbb Z)$ and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a nontrivial upper bound for the correlation $\sum _{n \leq X}f(n)\lambda _{\phi }(n+h)$ uniformly in $0<|h|\leq X$. As applications, we consider some special cases, including $f(n)=\lambda _{\pi }(n), \,\mu (n)\lambda _{\pi }(n)$ and any divisor-bounded multiplicative function. Here, $\lambda _{\pi }(n)$ denotes the $n$-th Dirichlet coefficient of a $\textrm{GL}_{m}$ automorphic $L$-function $L(s,\pi )$ for an irreducible cuspidal automorphic representation $\pi $, and $\mu (n)$ denotes the Möbius function. In particular, nontrivial savings are achieved for shifted convolution problems on $\textrm{GL}_{m}\times \textrm{GL}_{2}\, (m\geq 4)$ and Hypothesis C of Iwaniec–Luo–Sarnak for the first time.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Shandong Province
Young Scholars Program of Shandong University at Weiha
Taishan Scholars Project
National Key Research and Development Program of China
Publisher
Oxford University Press (OUP)