A lower bound for the least prime in an arithmetic progression

Junxian Li, Kyle Pratt, and I recently uploaded our paper A lower bound for the least prime in an arithmetic progression to the arXiv.

Here is a file where the heuristics considered in section 2 of the paper are developed in a slightly simpler situation.

Given a positive integer $k$ and a $\ell$ coprime to $k$, define $p(k , \ell)$ to be the smallest prime equivalent to $\ell$ modulo $k$. We are interested in the worst case behavior, that is $P(k) := \max_{(\ell , k) = 1} p(k , \ell).$ Thus $P(5) = 19$ and $P(12) = 13$. In particular we are interested in lower bounds for $P(k)$ for large $k$. An elementary observation, due to Pomerance, in Theorem 1 shows roughly that to find lower bounds for $P(k)$, it is enough to find lower bounds for the Jacobsthal function (the roughly will be explained below). For an integer $m$, the Jacobsthal function, $g(m)$, is the largest difference between consecutive integers coprime to $m$.

In recent work on Long gaps between primes by Ford, Green, Konyagin, Maynard, and Tao, they improve upon lower bounds for $g(m)$ where $m$ is the product of the first $u$ primes (they also mention the connection to the least prime problem; indeed it was Kevin Ford who originally introduced us to the problem). The key difference in the current problem is that we seek lower bounds for $g(m)$ where $m$ is the product of the first $u$ primes that are coprime to $k$. Our main new idea is to modify these sieve weights of Maynard used in the work of Ford, Green, Konyagin, Maynard, and Tao. We outline our approach in section 4 of our paper.

We finish by taking some time here to discuss smooth number estimates, which is perhaps the most important input to our work as well as all previous work on large gaps between primes (Westzynthius, in 1931, was the first to realize this connection). For $x \geq y \geq 1$, let $\Psi(x,y)$ be the number of integers $\leq x$ whose prime factors are all $\leq y$. Thus $\Psi(x,2)$ is the number of powers of $2$ that are at most $x$ and $\Psi(x , 3)$ is the number of integers of the form $2^a 3^b \leq x$. Estimating $\Psi(x,2)$ is straightforward and for $y$ is fixed, one can obtain an asymptotic for $\Psi(x,y)$ by counting lattice points in a simplex, as I describe in this previous blog post.

For our current problem, it is crucial that we are allowed to let $y$ depend on $x$. The important fact is that $\Psi(x,y)$ is much smaller than expected (by sieve theory heuristics). Rankin, in 1938, in his work on gaps between primes (see also: these set of notes) improved upon smooth number estimates to obtain better lower bounds for large gaps between primes. Westzynthius’ strategy, along with Rankin’s estimates, are still the starting points for current methods of constructing large gaps between primes.