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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Bounds for the first several prime character nonresidues
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by Paul Pollack PDF
Proc. Amer. Math. Soc. 145 (2017), 2815-2826 Request permission

Abstract:

Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon )$ and $\kappa =\kappa (\varepsilon ) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are more than $m^{\kappa }$ primes $\ell \le m^{\frac {1}{4\sqrt {e}}+\varepsilon }$ with $\chi (\ell )\notin \{0,1\}$. The proof uses the fundamental lemma of the sieve, Norton’s refinement of the Burgess bounds, and a result of Tenenbaum on the distribution of smooth numbers satisfying a coprimality condition. For quadratic characters, we demonstrate a somewhat weaker lower bound on the number of primes $\ell \le m^{\frac 14+\epsilon }$ with $\chi (\ell )=1$.
References
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Additional Information
  • Paul Pollack
  • Affiliation: Department of Mathematics, Boyd Graduate Studies Building, University of Georgia, Athens, Georgia 30602
  • MR Author ID: 830585
  • Email: pollack@uga.edu
  • Received by editor(s): August 24, 2015
  • Received by editor(s) in revised form: August 8, 2016
  • Published electronically: December 8, 2016
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2815-2826
  • MSC (2010): Primary 11A15; Secondary 11L40, 11N25
  • DOI: https://doi.org/10.1090/proc/13432
  • MathSciNet review: 3637932