Th\'eor\`eme de Chebotarev et complexit\'e de Littlewood

The effective version of Chebotarev's density theorem under the Generalized Riemann Hypothesis and the Artin conjecture (cf. Iwaniec and Kowalski's Analytic Number Theory, 5.13) involves a numerical invariant of a subset $D$ of a finite group $G$ that we call the Littlewood Complexity of $D$. We study this invariant in detail. Using this study, and a new application of the large sieve, we give improved versions of two standard questions related to Chebotarev: the bound on the first prime in a Frobenian set, and the asymptotic of the set of primes with given Frobenius in an infinite Galois extension. We then give concrete applications to various problems in number theory, such as the first primitive root modulo a prime $\ell$, the factorization of an integral polynomial modulo primes, the Lang-Trotter conjecture and its generalizations, and Serre's uniformity conjecture.