Bounds for the first several prime character nonresidues

Abstract

Let ε > 0 \varepsilon > 0 . We prove that there are constants m 0 = m 0 ( ε ) m_0=m_0(\varepsilon ) and κ = κ ( ε ) > 0 \kappa =\kappa (\varepsilon ) > 0 for which the following holds: For every integer m > m 0 m > m_0 and every nontrivial Dirichlet character modulo m m , there are more than m κ m^{\kappa } primes ℓ ≤ m 1 4 e + ε \ell \le m^{\frac {1}{4\sqrt {e}}+\varepsilon } with χ ( ℓ ) ∉ { 0 , 1 } \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 ℓ ≤ m 1 4 + ϵ \ell \le m^{\frac 14+\epsilon } with χ ( ℓ ) = 1 \chi (\ell )=1 .