Additive divisor problem for multiplicative functions

2022/04/18 by Yujiao Jiang, Guangshi Lü, Jiang, Yujiao +1
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.08221

openalex publication_date 2022/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ∑_{n ≤ 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 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 SL₂(\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.

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