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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 $p \rightarrow \infty$.

References [Enhancements On Off] (What's this?)

  • 1. Bundschuh, P., Solution of problem 618. Elemente der Mathematik 26 (1971), 43-44.
  • 2. Erdos, P., Bemerkungen zu einer Aufgabe in den Elementen. Arch. Math. 27 (1976), 159-163. MR 53:7969
  • 3. Erdos, P.; Murty, M. Ram, On the order of $a\pmod p$. Number theory (Ottawa, 1996), 87-97, CRM Proc. Lecture Notes, 19. MR 2000c:11152
  • 4. Gallagher, P.X., A larger sieve. Acta Arith. 18 (1971), 77-81. MR 45:214
  • 5. Hooley, C., On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209-220. MR 34:7445
  • 6. Matthews, C.R., Counting points modulo $p$ for some finitely generated subgroups of algebraic groups. Bull. London Math. Soc. 14 (1982), 149-154. MR 83c:10067
  • 7. Murty, M. Ram, Artin's conjecture for primitive roots. Math. Intelligencer 10 (1988), 59-67. MR 89k:11085
  • 8. Pappalardi, F., On the order of finitely generated subgroups of $Q^*(\operatorname{mod} p)$and divisors of $p-1$. J. Number Theory 57 (1996), 207-222. MR 97d:11141

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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