Large Gaps between Primes in Arithmetic Progressions

TL;DR

This paper establishes that the gaps between primes in specific arithmetic progressions can be arbitrarily large, with explicit lower bounds depending on the modulus and the size of the primes, extending understanding of prime distribution.

Contribution

It provides new lower bounds for the maximal gaps between primes in arithmetic progressions, generalizing previous results and applying to a broad class of moduli under certain conditions.

Findings

Gaps between primes in arithmetic progressions can grow arbitrarily large.

Explicit lower bounds depend on the modulus and logarithmic factors.

Results hold uniformly for moduli up to a specified size with controlled divisor structure.

Abstract

For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show that for any positive $C$, provided $X$ is large enough in terms of $C$, there holds \begin{equation*} G(MX;M,a)\geq(C+o(1))\varphi(M)\frac{\log X\log_2 X\log_4 X} {{(\log_3 X)}^2}, \end{equation*} uniformly for all $M\leq\kappa{(\log X)}^{1/5}$ that satisfy \begin{equation*} \omega(M)\leq \exp\biggl(\frac{\log_2 M\log_4 M}{\log_3 M}\biggr). \end{equation*}