Note on a sum involving the divisor function

TL;DR

This paper derives an improved asymptotic formula for a sum involving the divisor function and the integer part of a real expression, refining previous results by Feng.

Contribution

It provides a sharper asymptotic estimate for a divisor sum involving the floor function, advancing the understanding of divisor sums with fractional powers.

Findings

Established a new asymptotic formula with a smaller error term.

Improved upon Feng's previous results on divisor sums.

Derived explicit constants for the main term.

Abstract

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+\mathcal{O}_{\varepsilon,c} \left(x^{\max\{(2c+2)/(2c^2+5c+2),5/(5c+6)\}+\varepsilon}\right),$$ where $d_c=\sum_{k\geq1}d(k)\left(\frac{1}{k^{1/c}}-\frac{1}{(k+1)^{1/c}}\right)$ is a constant. This result constitutes an improvement upon that of Feng.