Distribution of elements of a floor function set in arithmetical progression

• Published in: arXiv.org
• Main Authors: Yu, Yahui, Wu, Jie
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 29.12.2021
• Subjects: Mathematics - Number Theory
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2112.14427

Let \([t]\) be the integral part of the real number \(t\).The aim of this short note is to study the distribution of elements of the set \(\mathcal{S}(x) := \{[\frac{x}{n}] : 1\le n\le x\}\) in the arithmetical progression \(\{a+dq\}_{d\ge 0}\).Our result is as follows: the asymptotic formula\begin{equation}\label{YW:result}S(x; q, a):= \sum_{\substack{m\in \mathcal{S}(x)\\ m\equiv a ({\rm mod}\,q)}} 1 = \frac{2\sqrt{x}}{q} + O((x/q)^{1/3}\log x)\end{equation}holds uniformly for \(x\ge 3\), \(1\le q\le x^{1/4}/(\log x)^{3/2}\) and \(1\le a\le q\),where the implied constant is absolute.The special case of \eqref{YW:result} with fixed \(q\) and \(a=q\) confirms a recent numeric test of Heyman.