Additive divisor problem for multiplicative functions

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 ${\sum }_{n\le X}f(n)\tau (n-1)$. We also derive several applications to multiplicative functions in the automorphic context, including the functions ${\lambda }_{\pi }(n),\mu (n){\lambda }_{\pi }(n)$ and ${\lambda }_{\varphi }(n{)}^l$. 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 $\pi$, ${\lambda }_{\varphi }(n)$ denotes the $n$-th Fourier coefficient of a holomorphic or Maass cusp form $\varphi$ on ${\mathrm{SL}}_2(\mathbb{Z})$, and $\mu (n)$ denotes the Möbius 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-Kátai-Sarnak-Ziegler criterion and Linnik's dispersion method are both employed in these two arguments.