Medium-sized values for the Prime Number Theorem for primes in arithmetic progression

TL;DR

This paper presents two improved explicit versions of the prime number theorem for primes in arithmetic progression, focusing on medium-sized values and the contribution of Siegel zeros, leading to better explicit results similar to Bombieri-Vinogradov.

Contribution

It introduces two new explicit bounds for primes in arithmetic progressions, one isolating Siegel zero effects and the other fully explicit for medium-sized values.

Findings

Improved explicit bounds for primes in arithmetic progressions.

Enhanced Bombieri-Vinogradov type results for non-exceptional moduli.

Explicit formulas accounting for Siegel zero contributions.

Abstract

We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized values. This will give an improved explicit Bombieri-Vinogradov like result for non-exceptional moduli.