On short products of primes in arithmetic progressions

Shparlinski, Igor

doi:10.1090/proc/14289

On short products of primes in arithmetic progressions
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by Igor E. Shparlinski PDF

Proc. Amer. Math. Soc. 147 (2019), 977-986 Request permission

Abstract:

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha , \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots , p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at most $\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdős, A. M. Odlyzko and A. Sárközy (1987), that corresponds to $k = \ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).

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