Primes in arithmetic progressions on average I

TL;DR

This paper investigates the distribution of primes in arithmetic progressions, demonstrating that sign changes in the error term lead to significant cancellation effects, surpassing traditional heuristic expectations.

Contribution

It introduces a new bound on the sum of the cube of error terms, revealing deeper cancellation phenomena in primes in arithmetic progressions.

Findings

Established a bound: M(Q) << Q^3 (x/Q)^{7/5} for large Q

Showed sign changes in error terms cause power-saving cancellation

Surpassed the usual sqrt(x/q) heuristic expectation

Abstract

Let $E_x(q,a)$ be the error term when counting primes in arithmetic progressions and let $M(Q)=\sum_{q\leq Q}\phi(q)\sum_{a=1}^qE_x(q,a)^3$. We show that $M(Q)<<Q^3(x/Q)^{7/5}$ for large $Q$ close to $x$ (in the usual BDH sense) thereby showing that sign changes in the error give power saving cancellation past the expected $\sqrt {x/q}$ heuristic.