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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The distribution of sequences in residue classes
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by Christian Elsholtz PDF
Proc. Amer. Math. Soc. 130 (2002), 2247-2250 Request permission


We prove that any set of integers ${\mathcal A}\subset [1,x]$ with $\vert {\mathcal A} \vert \gg (\log x)^r$ lies in at least $\nu _{\mathcal A}(p) \gg p^{\frac {r}{r+1}}$ many residue classes modulo most primes $p \ll (\log x)^{r+1}$. (Here $r$ is a positive constant.) This generalizes a result of Erdős and Ram Murty, who proved in connection with Artin’s conjecture on primitive roots that the integers below $x$ which are multiplicatively generated by the coprime integers $a_1, \ldots , a_r$ (i.e. whose counting function is also $c ( \log x)^r$) lie in at least $p^{\frac {r}{r+1} + \varepsilon (p)}$ residue classes, modulo most small primes $p$, where $\varepsilon (p) \rightarrow 0,$ as $p \rightarrow \infty$.
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Additional Information
  • Christian Elsholtz
  • Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany
  • Email:
  • Received by editor(s): March 9, 2001
  • Published electronically: January 23, 2002
  • Communicated by: David E. Rohrlich
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 130 (2002), 2247-2250
  • MSC (1991): Primary 11N69, 11N36; Secondary 11B50, 11A07
  • DOI:
  • MathSciNet review: 1896404