An explicit version of Bombieri's log-free density estimate

Jesse Thorner, Asif Zaman

Published 2022-08-23Version 1

We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of S{\'a}rk{\"o}zy. Also, for integers $q\geq 3$ and $a$ such that $\gcd(a,q)=1$, we prove a highly uniform and explicit version of the prime number theorem that nontrivially counts primes $p\leq x$ such that $p\equiv a\pmod{q}$ when $x\geq \max\{q,411\}^{150\,000}$.