Note on fractional sum of the divisor function

Zhiyuan Yang

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arXiv · 2610.04579

Note on fractional sum of the divisor function

Zhiyuan Yang

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Abstract

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{θ_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $θ_c<2/(3c+2)$ for $0<c<2/11$. This result constitutes an improvement upon that of Y. Feng (2024).

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Zhiyuan Yang. 2026-10-08. Note on fractional sum of the divisor function. https://arxiv.org/abs/2610.04579

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