Chebyshev's Bias against Splitting and Principal Primes in Global Fields

Abstract

A reason for the emergence of Chebyshev's bias is investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for making a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, we succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which gives a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $\sigma$ in the Galois group, we establish a criterion of the bias of primes whose Frobenius elements are equal to $\sigma$ under the assumption of DRH. As an application we obtain a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is proved, and all these results hold unconditionally