On Higher-Power Moments of $ Δ_a(x) $ for $-1/2<a<0$
Yi Cai, Jinjiang Li, Yankun Sui, Fei Xue, Min Zhang
TL;DR
This paper studies higher-power moments of the generalized error term $\Delta_a(x)$ associated with the divisor function $\sigma_a(n)$ for $-\tfrac{1}{2}<a<0$, aiming to understand the magnitude of $\Delta_a(x)$ through mean values. The authors employ a truncated Voronoï-type decomposition to separate a main oscillatory term from a remainder, deriving an explicit main-term coefficient $C_k(a)$ and an optimal error bound for $3\le k\le 7$. Their results extend Zhai's bounds to $k\le 7$ (with improved constants and explicit delta-exponents) and provide a conditional pathway to larger $k$ contingent on stronger bounds for $\Delta_a(x)$. The methods combine oscillatory integral estimates, near-equality counting for square-root sums, and large-value $L^{A_0}$-mean bounds, contributing to the broader understanding of divisor-problem-type moments and their asymptotics.
Abstract
Let $-1/2<a<0$ be a fixed real number and \begin{equation*} Δ_{a}(x)=\sideset{}{'}\sum_{n\leq x} σ_a(n)-ζ(1-a)x-\frac{ζ(1+a)}{1+a}x^{1+a}+\frac{1}{2}ζ(-a). \end{equation*} In this paper, we investigate the higher--power moments of $Δ_a(x)$ and give the corresponding asymptotic formula for the integral $\int_{1}^{T}Δ_a^k(x)\mathrm{d}x$, 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 $7$.