Chebyshev's bias and the Chebotarev density theorem

A well known phenomenon in prime number theory is the so called "Chebyshev bias": it describes the predominance, for "most" real numbers x> 0, of the number of primes < x and congruent to 3 modulo 4 over primes < x and congruent to 1 modulo 4. In the 90's, Rubinstein and Sarnak have given a very general framework for natural generalizations of Chebyshev's bias. They noably emphasized the role played by the zeros of the relevant L-functions. In particular their potential property of linear independence on Q is of crucial importance. I will present a work in progress, in common with Daniel Fiorilli, in which we study a variant of Chebyshev's question for general Galois extensions of number fields. In this context the prime number theorem in arithmetic progressions is replaced by Chebotarev's theorem. We will see on the one hand that there is "generically no bias" and on the other hand, we will describe families of number fields giving rise to extreme biases.