On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli

Abstract

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Teräväinen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^{\gamma }$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_{\bullet }\le q$ for which $q_{\bullet }\ge \exp (C(\log q{)}^{2/3}(\log \log q{)}^{4/3})$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_{\bullet }$. We also show that the conjecture holds for all but a proportion $\exp (-c(\log q{)}^{1/3}(\log \log q{)}^{-1/3})$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^{\gamma }$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.