Unconditional Chebyshev biases in number fields

TL;DR

This paper demonstrates unconditionally the existence of extreme Chebyshev biases in number fields, showing certain prime distributions always favor one conjugacy class over another, beyond previous theoretical predictions.

Contribution

It proves unconditionally that extreme biases in prime distributions exist in number fields, bypassing the need for unproven hypotheses about zeros of Artin L-functions.

Findings

Existence of infinite families of Galois extensions with biased prime distributions

Biases are larger than those predicted by effective Chebotarev bounds

Circumvents reliance on unproven hypotheses about zeros of Artin L-functions

Abstract

Prime counting functions are believed to exhibit, in various contexts, discrepancies beyond what famous equidistribution results predict; this phenomenon is known as Chebyshev's bias. Rubinstein and Sarnak have developed a framework which allows to conditionally quantify biases in the distribution of primes in general arithmetic progressions. Their analysis has been generalized by Ng to the context of the Chebotarev density theorem, under the assumption of the Artin holomorphy conjecture, the Generalized Riemann Hypothesis, as well as a linear independence hypothesis on the zeros of Artin $L$-functions. In this paper we show unconditionally the occurrence of extreme biases in this context. These biases lie far beyond what the strongest effective forms of the Chebotarev density theorem can predict. More precisely, we prove the existence of an infinite family of Galois extensions and associated conjugacy classes $C_1,C_2\subset {\rm Gal}(L/K)$ of same size such that the number of prime ideals of norm up to $x$ with Frobenius conjugacy class $C_1$ always exceeds that of Frobenius conjugacy class $C_2$, for every large enough $x$. A key argument in our proof relies on features of certain subgroups of symmetric groups which enable us to circumvent the need for unproven properties of zeros of Artin $L$-functions.