On the second smallest prime non-residue

Journal of Number Theory (Impact Factor: 0.59). 11/2010; 133(4). DOI: 10.1016/j.jnt.2012.09.011
Source: arXiv


Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime non-residues of $\chi$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good upper estimate on the product $q_1 q_2$ which has an upcoming application to the study of norm-Euclidean Galois fields.