## Bounds for the first several prime character nonresidues

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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 $\chi (\ell )=1$.## 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