Distribution of elements of a floor function set in arithmetical progression

TL;DR

This paper investigates how the set of integer parts of x divided by n distributes across arithmetic progressions, providing an asymptotic formula that confirms recent numerical observations.

Contribution

The paper derives a uniform asymptotic formula for the distribution of floor(x/n) in arithmetic progressions, extending understanding of this set’s distribution.

Findings

Asymptotic formula for distribution in progressions

Uniform results for a range of q and a

Confirmation of recent numerical tests

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.