Refinements for primes in short arithmetic progressions

TL;DR

This paper improves error estimates in the prime number theorem for short arithmetic progressions under certain hypotheses, refining classical interval bounds for prime distribution.

Contribution

It introduces refined error terms for primes in short arithmetic progressions assuming zero-free regions and zero-density estimates, including the Generalized Density Hypothesis.

Findings

Prime number theorem holds in shorter intervals under new assumptions.

Refined bounds apply to all and almost all short intervals.

Results extend classical bounds for prime distribution in progressions.

Abstract

Given a zero-free region and an averaged zero-density estimate over all Dirichlet $L$-functions modulo $q\in\mathbb{N}$, we refine the error terms of the prime number theorem in all and almost all short arithmetic progressions. For example, if we assume the Generalized Density Hypothesis, then for any arithmetic progression modulo $q\leq \log^{\ell} x$ with $\ell>0$ and any $\varepsilon>0$, the prime number theorem holds in all intervals $(x-\sqrt{x}\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ and almost all intervals $(x-\exp(\log^{\frac{2}{3}+\varepsilon} x),x]$ as $x\rightarrow\infty$. This refines the classic intervals $(x-x^{1/2+\varepsilon},x]$ and $(x-x^\varepsilon,x]$ for any $\varepsilon>0$.