A Weighted Divisor Problem

TL;DR

This paper investigates a weighted divisor function involving trigonometric weights and rational parameters, establishing bounds and moments by linking it to the divisor problem with congruence conditions.

Contribution

It introduces a novel approach connecting a weighted divisor sum with the divisor problem under congruence conditions, providing new bounds and moment estimates.

Findings

Established upper bounds for the weighted divisor function

Derived mean-value and mean-square estimates

Analyzed power-moments of the function

Abstract

We study a weighted divisor function $\mathop{{\sum}'}\limits_{mn\leq x}\cos(2\pi m\theta_1)\sin(2\pi n\theta_2)$, where $\theta_i (0<\theta_i<1)$ is a rational number. By connecting it with the divisor problem with congruence conditions, we establish the upper bound, mean-value, mean-square and some power-moments.