Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

Abstract

Let $q\ge 3$ be a non-exceptional modulus $q\ge 3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon >0$, there exists $\alpha >0$ (computable), such that for all $x\ge \alpha (\log q{)}^2$, the interval $\left[e^x,e^{x+\epsilon }\right]$ contains a prime $p$ in the arithmetic progression $a\mathrm{mod}q$. This gives the bound for the least prime in this arithmetic progression: $P(a,q)\le e^{\alpha (\log q{)}^2}$. For instance for all $q\ge 10^{30}$, $P(a,q)\le e^{4.401(\log q{)}^2}$. Finally, we apply this result to establish that every integer larger than $e^{71000}$ is a sum of seven cubes.