On short products of primes in arithmetic progressions

2017/05/17 by Igor E. Shparlinski, Shparlinski, Igor E.
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1705.06087

openalex publication_date 2017/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give several families of reasonably small integers k, ℓ ≥ 1 and real positive α, β≤ 1, such that the products p₁… p_k s, where p₁, …, p_k ≤ m^α are primes and s ≤ m^β is a product of at most ℓ primes, represent all reduced residue classes modulo m. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to k = ℓ =1 (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).

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