A Bombieri–Vinogradov-type theorem with prime power moduli
Volume 204 / 2022
Abstract
In 2020, Roger Baker 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 of 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 $C \gt 0$. 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 (1964). Gallagher (1972) extended this range to $p^N\le x^{2/5-\varepsilon }$, and Huxley (1975) improved Gallagher’s exponent to $5/12$. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski (2019) for the even wider range $p^N\le x^{0.4736}$. However, all these results hold for 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.
