On moments of the error term of the multivariable k-th divisor functions

2024/11/11 by Zhen Guo, Xin Li, Guo, Zhen +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.06656

openalex publication_date 2024/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose k\geqslant3 is an integer. Let τ_k(n) be the number of ways n can be written as a product of k fixed factors. For any fixed integer r\geqslant2, we have the asymptotic formula ∑_{n1,⋯,nr\leqslant x}τ_k(n₁ ⋯ n_r)=x^r∑_ℓ=0^r(k-1)d_r,k,ℓ(log x)^ℓ+O(x^r-1+α_k+ε), where d_r,k,ℓ and 0<α_k<1 are computable constants. In this paper we study the mean square of Δ_r,k(x) and give upper bounds for k\geqslant4 and an asymptotic formula for the mean square of Δ_r,3(x). We also get an upper bound for the third power moment of Δ_r,3(x). Moreover, we study the first power moment of Δ_r,3(x) and then give a result for the sign changes of it.

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