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.
Categories: math.NT