Correlations of multiplicative functions with automorphic L-functions

Let $λ_ϕ(n)$ 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 $\sum_{n \leq X}f(n)λ_ϕ(n+h)$ uniformly in $0<|h|\ll X$. As applications, we consider some special cases, including $λ_π(n), \,μ(n)λ_π(n)$ and any divisor-bounded multiplicative function. Here $λ_π(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,π)$ for an automorphic irreducible cuspidal representation $π$, and $μ(n)$ denotes the Möbius 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.

No reviews yet.

No comments yet.