论文标题
乘法功能与自动型L功能的相关性
Correlations of multiplicative functions with automorphic L-functions
论文作者
论文摘要
令$ λ_ϕ（n）$为$ {\ rm sl} _2（\ mathbb z）$上的Hecke Holomorphic或Hecke-maass cusp表格的傅立叶系数，而$ f $是任何满足两个轻度假设的多重功能。我们建立了一个相关性的非平凡的上限$ \ sum_ {n \ leq x} f（n）λ_ϕ（n+h）$均匀地以$ 0 <| h | h | \ ll x $。作为应用程序，我们考虑了一些特殊情况，包括$λ_π（n），\，μ（n）λ_π（n）$和任何分裂的乘法函数。 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.特别是，在$ {\ rm gl} _m \ times {\ rm gl} _2 \，（m \ geq 4）$和假设C上，第一次节省了一些节省的卷积问题。
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.
