A Bombieri-Vinogradov-type theorem with prime power moduli

Abstract

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arbitrary $C>0$. For large enough $C$, we thus get an almost all result. Previously, an asymptotic estimate for $\pi(x;p^N,a)$ of the expected kind, with $p$ being an odd prime, was established in the wider range $p^N\le x^{3/8-\varepsilon}$ by Barban, Linnik and Chudakov \cite{BLC}. Gallagher \cite{Gal} extended this range to $p^N\le x^{2/5-\varepsilon}$ and Huxley \cite{Hux2} improved Gallagher's exponent to $5/12$. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski \cite{BaS} for the even wider range $p^N\le x^{0.4736}$. However, all these results hold for {\it fixed} primes $p$, and the $O$-constants in the relevant estimates depend on $p$. Therefore, they do not contain our result. In a part of our article, we describe how our method relates to these results.