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

Abstract

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, 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 $a \bmod q$. 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 $e^{71\,000}$ is a sum of seven cubes.