Additive divisor problem for multiplicative functions

TL;DR

This paper investigates the additive divisor problem for multiplicative functions, establishing asymptotic formulas and bounds for shifted convolution sums, with applications to automorphic forms and novel proof techniques.

Contribution

It introduces new methods to analyze shifted convolution sums involving multiplicative functions, applying recent estimates and classical techniques in analytic number theory.

Findings

Established asymptotic formulas for shifted sums involving multiplicative functions.

Derived bounds and applications to automorphic form coefficients.

Presented two distinct proof strategies using recent advances and classical methods.

Abstract

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq X}f(n)\tau(n-1)$. We also derive several applications to multiplicative functions in the automorphic context, including the functions $\lambda_{\pi}(n), \,\mu(n)\lambda_{\pi}(n)$ and $\lambda_{\phi}(n)^l$. 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 $\pi$, $\lambda_{\phi}(n)$ denotes the $n$-th Fourier coefficient of a holomorphic or Maass cusp form $\phi$ on ${\rm SL}_2(\mathbb Z)$, and $\mu(n)$ denotes the M\"obius function. We present two different arguments. The first one mainly relies on the uniform estimates for the binary additive divisor problem, while the second is based on the recent estimates of Bettin--Chandee for trilinear forms in Kloosterman fractions. In addition, the Bourgain-K\'atai-Sarnak-Ziegler criterion and Linnik's dispersion method are both employed in these two arguments.