Bounds for the first several prime character nonresidues

TL;DR

This paper establishes lower bounds on the number of prime nonresidues for Dirichlet characters within certain bounds, using advanced sieve methods and bounds on smooth numbers, advancing understanding of character nonresidues.

Contribution

It provides new quantitative bounds on the distribution of prime nonresidues for Dirichlet characters, extending previous results with explicit constants and refined techniques.

Findings

More than m^{ppa} primes are nonresidues within specified bounds.

For quadratic characters, a lower bound on primes with hi(\u00aell)=1 is established.

The results depend on advanced sieve methods and bounds on smooth numbers.

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 $\ell \le m^{\frac14+\epsilon}$ with $\chi(\ell)=1$.