A lower bound for the least prime in an arithmetic progression

TL;DR

This paper establishes a new lower bound for the least prime in an arithmetic progression for most moduli, using sieve weights and recent results on prime gaps, suggesting the bound is close to optimal.

Contribution

It introduces a novel approach using sieve weights to improve lower bounds for the least prime in arithmetic progressions, answering a longstanding question.

Findings

For almost all k, P(k) is significantly larger than previously known bounds.

The method captures small multiples of primes, not just primes.

Heuristic suggests P(k) / (φ(k) log^2 k) approaches 1 as k grows.

Abstract

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg \phi(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ \phi(k) \log^2 k} = 1.$