论文标题
乘法功能的加性除数问题
Additive divisor problem for multiplicative functions
论文作者
论文摘要
令$τ$表示除数函数，而$ f $是满足某些温和假设的任何乘法功能。我们为移位卷积总和$ \ sum_ {n \ leq x} f（n）τ（n-1）$建立了渐近公式或非平凡的上限。我们还将多个应用程序推导到自动形上下文中的乘法函数，包括$λ_π（n），\，μ（n）λ_π（n）$和$λ_Dartive（n）^l $。这里$λ_π（n）$表示$ n $ th dirichlet系数的$ \ text {gl} _m $ $ $ $ usormorthic $ l $ l $ -function $ l（s，π）$用于自动形态不可差的cuspidal cuspidal代表$π$π$，$λ__（$λ_（ $ {\ rm sl} _2（\ mathbb z）$上的cusp form $ ϕ $，$μ（n）$表示möbius函数。 我们提出两个不同的论点。第一个主要依赖于二进制添加剂问题的均匀估计值，而第二个是基于贝丁（Bettin）的最新估计值 - 贝特（Kloosterman）分数中三线性形式的chandee。此外，这两个论点都采用了Bourgain-Kátai-Sarnak-Ziegler标准和Linnik的分散方法。
Let $τ$ 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 $\sum_{n \leq X}f(n)τ(n-1)$. We also derive several applications to multiplicative functions in the automorphic context, including the functions $λ_π(n), \,μ(n)λ_π(n)$ and $λ_ϕ(n)^l$. 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 $π$, $λ_ϕ(n)$ denotes the $n$-th Fourier coefficient of a holomorphic or Maass cusp form $ϕ$ on ${\rm SL}_2(\mathbb Z)$, and $μ(n)$ 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.
