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
Abstract:
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 $p \rightarrow \infty$.References
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Additional Information
- Christian Elsholtz
- Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany
- Email: elsholtz@math.tu-clausthal.de
- 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: https://doi.org/10.1090/S0002-9939-02-06395-5
- MathSciNet review: 1896404