Primes in short arithmetic progressions

TL;DR

This paper demonstrates that for most moduli and intervals, primes are evenly distributed in short arithmetic progressions under certain size conditions, advancing understanding of prime distribution in such progressions.

Contribution

It establishes new bounds for the distribution of primes in short arithmetic progressions for most moduli and intervals, extending previous results with refined conditions.

Findings

Primes are evenly distributed in most short arithmetic progressions under specified bounds.

Most moduli up to Q have at least one prime in the interval in every reduced residue class.

New bounds relate the length of intervals, moduli size, and prime distribution in progressions.

Abstract

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+\epsilon}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod q$, provided that $1\le Q^2\le h/x^{1/15+\epsilon}$.