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 $q=p^γ$ 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 $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, 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 $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ 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 $p^γ$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.