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Bounds for the first several prime character nonresidues


Author: Paul Pollack
Journal: Proc. Amer. Math. Soc. 145 (2017), 2815-2826
MSC (2010): Primary 11A15; Secondary 11L40, 11N25
DOI: https://doi.org/10.1090/proc/13432
Published electronically: December 8, 2016
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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$.


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Additional Information

Paul Pollack
Affiliation: Department of Mathematics, Boyd Graduate Studies Building, University of Georgia, Athens, Georgia 30602
Email: pollack@uga.edu

DOI: https://doi.org/10.1090/proc/13432
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
Article copyright: © Copyright 2016 American Mathematical Society