Distribution of elements of a floor function set in arithmetical progression

Abstract

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.