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

TL;DR

This paper investigates the moments of the error term in the asymptotic formula for multivariable divisor functions, providing bounds and formulas for different cases, and analyzing their sign changes.

Contribution

It offers new bounds and asymptotic formulas for the moments of the error term of multivariable divisor functions, especially for the case when k=3.

Findings

Upper bounds for the mean square of the error term for k≥4

Asymptotic formula for the mean square when k=3

Bounds for the third power moment when k=3

Abstract

Suppose $k\geqslant3$ is an integer. Let $\tau_k(n)$ 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 \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}\tau_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+\alpha_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<\alpha_k<1$ are computable constants. In this paper we study the mean square of $\Delta_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $\Delta_{r,3}(x)$. We also get an upper bound for the third power moment of $\Delta_{r,3}(x)$. Moreover, we study the first power moment of $\Delta_{r,3}(x)$ and then give a result for the sign changes of it.