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The distribution of sequences in residue classes

Authors:	Christian Elsholtz

Institution:	Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany

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 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 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 , where $\varepsilon(p) \rightarrow 0,$ as $p \rightarrow \infty$ .

Keywords:	Distribution of sequences in residue classes Gallagher's larger sieve primitive roots Artin's conjecture

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