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=pγ that are powers of a fixed prime p and we show that the third prime can be taken subpolynomially small. For any q•≤ q for which q•≥exp(C(log q)2/3(loglog 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•. We also show that the conjecture holds for all but a proportion exp(−c(log q)1/3(loglog 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 pγ.We also present conditional versions of these results for arbitrary q under a quasi-Riemann hypothesis.
Related
- Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
- Prime Numbers of the Form 24m+1 and Erdös-Straus Conjecture
- A negative answer to the Erdős-Sárkőzy question
- A solution to the Erdős-Sárközy product-density problem
- The Odlyzko-Poonen Conjecture on Irreducibility of Random Polynomials
BibTeX
@misc{banks2026erdősodlyzkosárközy,
title = {{On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli}},
author = {William Banks and Igor E. Shparlinski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19597},
eprint = {2610.19597},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19597},
}
RIS
TY - UNPB
TI - On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli
AU - William Banks
AU - Igor E. Shparlinski
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19597
UR - https://arxiv.org/abs/2610.19597
AB - 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_•≤ q for which q_•≥exp(C(log q)^2/3(loglog 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_•. We also show that the conjecture holds for all but a proportion exp(−c(log q)^1/3(loglog 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^γ.We also present conditional versions of these results for arbitrary q under a quasi-Riemann hypothesis.
ER -