Correlations of multiplicative functions with automorphic L-functions

Abstract

Let ${\lambda }_{\varphi }(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\mathrm{SL}}_2(\mathbb{Z})$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper bound for the correlation ${\sum }_{n\le X}f(n){\lambda }_{\varphi }(n+h)$ 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 Möbius function. In particular, some savings are achieved for shifted convolution problems on ${\mathrm{GL}}_m\times {\mathrm{GL}}_2(m\ge 4)$ and Hypothesis C for the first time.