Correlations of multiplicative functions with automorphic L-functions

TL;DR

This paper establishes bounds on correlations between multiplicative functions and automorphic L-function coefficients, advancing understanding of shifted convolution problems and achieving savings under Hypothesis C.

Contribution

It provides the first non-trivial bounds for correlations involving automorphic L-function coefficients and multiplicative functions, including cases with shifts and divisor-bounded functions.

Findings

Established uniform upper bounds for correlations with shifts.

Achieved savings in shifted convolution problems on GL_m x GL_2.

Provided new results under Hypothesis C for automorphic forms.

Abstract

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper bound for the correlation $\sum_{n \leq X}f(n)\lambda_{\phi}(n+h)$ uniformly in $0<|h|\ll X$. As applications, we consider some special cases, including $\lambda_{\pi}(n), \,\mu(n)\lambda_{\pi}(n)$ and any divisor-bounded multiplicative function. Here $\lambda_{\pi}(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,\pi)$ for an automorphic irreducible cuspidal representation $\pi$, and $\mu(n)$ denotes the M\"obius function. In particular, some savings are achieved for shifted convolution problems on ${\rm GL}_m\times {\rm GL}_2\, (m\geq 4)$ and Hypothesis C for the first time.