Bounds for the first several prime character nonresidues

Abstract

Let $\epsilon >0$. We prove that there are constants $m_0=m_0(\epsilon )$ and $\hat{I}º=\hat{I}º(\epsilon )>0$ for which the following holds: For every integer $m>m_0$ and every nontrivial Dirichlet character modulo $m$, there are more than $m^{\hat{I}}º$ primes $\ell \le m^{\frac{1}{4\sqrt{e}}+\epsilon }$ with $\ddot{I}‡(\ell )\notin \{0,1\}$. 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 $\ell \le m^{\frac{1}{4}+\hat{I}\mu }$ with $\ddot{I}‡(\ell )=1$.

Theorem 1.3 has been removed, as the same result (with the same proof) already appears in work of Aled Walker; see Lemma 9 of http://arxiv.org/abs/1505.03328v3