The prime number theorem for primes in arithmetic progressions at large values

TL;DR

Under the assumption of the Riemann hypothesis, the paper derives explicit formulas and theorems related to the distribution of primes in arithmetic progressions, including refined results for moduli up to 10,000.

Contribution

It provides the latest explicit versions of the prime number theorem for short intervals and primes in arithmetic progressions under GRH, with computational refinements for small moduli.

Findings

Explicit formulas for $ heta(x, ext{chi})$ and $ ext{psi}(x, ext{chi})$

Refined results for moduli $q \,\leq 10,000$

Conditional proofs assuming GRH and RH

Abstract

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish explicit formulae for $\psi(x,\chi)$, $\theta(x,\chi)$, and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.