On Higher-Power Moments of $ \Delta_a(x) $ for $-1/2<a<0$

TL;DR

This paper studies the higher-power moments of the function (x) related to divisor sums, providing new asymptotic formulas and extending previous bounds for moments up to the seventh power.

Contribution

It offers improved asymptotic formulas for the moments of (x) and extends the known upper bounds for the power k up to 7.

Findings

Asymptotic formulas for (x) moments for k=3,4,5

Extended upper bound of k to 7

Improved understanding of divisor sum fluctuations

Abstract

Let $-1/2<a<0$ be a fixed real number and \begin{equation*} \Delta_{a}(x)=\sideset{}{'}\sum_{n\leq x} \sigma_a(n)-\zeta(1-a)x-\frac{\zeta(1+a)}{1+a}x^{1+a}+\frac{1}{2}\zeta(-a). \end{equation*} In this paper, we investigate the higher--power moments of $\Delta_a(x)$ and give the corresponding asymptotic formula for the integral $\int_{1}^{T}\Delta_a^k(x)\mathrm{d}x$, which constitutes an improvement upon the previous result of Zhai [9] for $k=3,4,5$ and an enlargement of the upper bound of $k$ to $7$.