On short products of primes in arithmetic progressions

Shparlinski, Igor E.

doi:10.1090/proc/14289

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On short products of primes in arithmetic progressions

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Shparlinski, Igor E.
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1090/proc/14289
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arXiv:1705.06087.pdf
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Shparlinski, I. E. (2019). On short products of primes in arithmetic progressions. Proceedings of the American Mathematical Society, 147(3), 977-986. doi:10.1090/proc/14289.

Cite as: https://hdl.handle.net/21.11116/0000-0004-F813-F

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. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \ell =1$ (that is, to
products of two primes). In particular, we improve recent results of A. Walker (2016).