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The distribution of sequences in residue classes

Author: Christian Elsholtz
Journal: Proc. Amer. Math. Soc. 130 (2002), 2247-2250
MSC (1991): Primary 11N69, 11N36; Secondary 11B50, 11A07
Published electronically: January 23, 2002
MathSciNet review: 1896404
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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 Erdos 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

Keywords: Distribution of sequences in residue classes, Gallagher's larger sieve, primitive roots, Artin's conjecture
Received by editor(s): March 9, 2001
Published electronically: January 23, 2002
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society

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