Bounds for the first several prime character nonresidues

2015/08/20 by Paul Pollack, Pollack, Paul
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1508.05035

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

arxiv created 2015/08/24 · arxiv updated 2015/08/25

Abstract

Let ε > 0. We prove that there are constants m₀=m₀(ε) and κ=κ(ε) > 0 for which the following holds: For every integer m > m₀ and every nontrivial Dirichlet character modulo m, there are more than m^κ primes ℓ ≤ m^{(1)/(4√(e))+ε} with χ(ℓ)∉ \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^\frac14+ε with χ(ℓ)=1.

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