On Vaughan's approximation: The first moment

TL;DR

This paper analyzes Vaughan's approximation for the first moment of prime distribution discrepancies in arithmetic progressions, demonstrating its superior accuracy over classical estimates within a specific range of q.

Contribution

It introduces a refined approximation that better captures prime distribution discrepancies, improving upon classical methods for the first moment analysis.

Findings

Vaughan's approximation outperforms classical x/φ(q) in accuracy.

The approximation effectively captures known prime distribution discrepancies.

Results are valid within a specific range of q.

Abstract

We investigate the first moment of the difference between $\psi(x;q,a)$ and Vaughan's approximation, in a certain range of $q$. We show that this last approximation is significantly more precise than the classical $x/\phi(q)$, and that it captures the discrepancies of the distribution of primes in arithmetic progressions found in an earlier paper of the author.