Primes in arithmetic progressions on average I

Tomos Parry

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arXiv · 2406.06450

Primes in arithmetic progressions on average I

Tomos Parry

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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.

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Tomos Parry. 2024-06-10. Primes in arithmetic progressions on average I. https://arxiv.org/abs/2406.06450

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