Higher order divisor functions over values of mixed powers

TL;DR

This paper derives an asymptotic formula for sums involving higher order divisor functions over mixed power sums, extending previous work limited to special cases, and advancing understanding of divisor functions in complex sum structures.

Contribution

The paper introduces a general asymptotic formula for sums of divisor functions over mixed powers, broadening the scope beyond previously studied special cases.

Findings

Established an asymptotic formula for the sum involving divisor functions over mixed powers.

Extended the analysis to general parameters r, s, k, and , surpassing prior specific case studies.

Provides new tools for analyzing divisor functions in complex summation contexts.

Abstract

Let $\tau_k(n)$ be the $k$-th divisor function. In this paper, we derive an asymptotic formula for the sum $$ \sum_{1\leq n_1,n_2, \dots, n_{\ell}\leq X^{\frac{1}{r}} \atop 1\leq n_{\ell+1}\le X^{\frac{1}{s}}}\tau_k(n_1^r+n_2^r+\dots +n_{\ell}^r+n_{\ell+1}^s), $$ where $k\geq 4$, $r\geq 2$, $s\geq 2$ and $\ell\geq 2$ are integers. Previously only special cases are studied.