Products in Residue Classes

TL;DR

This paper investigates the representation of residue classes modulo m as products of two primes, providing unconditional average results and stronger findings for prime moduli, with extensions to easier sequences.

Contribution

It offers new unconditional average results on the representation of residue classes by prime products, surpassing previous limitations under the Extended Riemann Hypothesis.

Findings

Unconditional average results over moduli m and residue classes

Stronger results for prime moduli m=p

Enhanced results when primes are replaced by easier sequences

Abstract

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.