# Explicit bounds for small prime nonresidues

## TL;DR

This paper provides explicit upper bounds for the smallest prime nonresidues of Dirichlet characters modulo a prime, improving understanding of their distribution with precise bounds depending on the prime and the number of nonresidues.

## Contribution

It establishes explicit bounds for the smallest prime nonresidues of Dirichlet characters, with constants depending on given parameters, advancing previous theoretical results.

## Key findings

- Bounds depend on prime p and number of nonresidues n
- Explicit constants are provided for practical bounds
- Results hold for sufficiently large p and fixed n

## Abstract

Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant $C=C(n_0,p_0)>0$ such that $q_n\leq Cp^{\frac{1}{4}}(\log p)^{\frac{n+1}{2}}$ whenever $n\leq n_0$ and $p\geq p_0$.