Primes in the Chebotarev density theorem for all number fields

Abstract

We establish an explicit bound for the least prime occurring in the Chebotarev density theorem without any restriction. Let L/K be any Galois extension of number fields such that L = Q, and let C be a conjugacy class in the Galois group of L/K. We show that there exists an unramified prime p of K such that σp = C and N p ≤ d B L with B = 310. This improves the value B = 12 577 as proven by Ahn and Kwon. In comparison to previous works on the subject, we modify the weights to detect the least prime, and we use a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn (zero-repulsion) phenomenon. In addition, we refine the analysis of how the location of the potential exceptional zero for ζL(s) affects the final result. We also use Fiori's numerical verification for L up to a certain discriminant height. Finally, we provide a lower bound for the number of unramified primes p of K such that σp = C.

Uniform explicit Stewart's theorem on prime factors of linear recurrences

“…For instance, Theorem 5 of [7] applies for x ≥ x 1 := exp(|D L | 12 ), where, in our case, L is the Hilbert Class Field of K. We have |D L | = |D K | hK , which would lead to a double exponential value for x 1 . And p 0 , as it is clear from the proof of Theorem 1.5, is exponential in x 1 , leading to the triple exp dependence of p 0 in |D K |.…”

“…Using the Class Field Theory and the Tchebotarev Density Theorem, one can show that the relative density of Stewart primes in the set of all primes is (2h K ) −1 . Moreover, using recent explicit versions of the the Tchebotarev Density Theorem, as in [1,7,13], one can give a totally explicit lower estimate for the counting function of Stewart primes.…”