Bounds for the first several prime character nonresidues

Pollack, Paul

doi:10.1090/proc/13432

Bounds for the first several prime character nonresidues
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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 $\chi (\ell )=1$.

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