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

TL;DR

This paper explores Chebyshev's bias in global fields, linking it to the Deep Riemann Hypothesis (DRH), and provides new asymptotic formulas and criteria for prime distribution biases, especially in abelian extensions.

Contribution

It introduces a new formulation of Chebyshev's bias using weighted prime counting and establishes criteria for prime biases in Galois extensions under DRH.

Findings

Bias towards non-splitting and non-principal primes in abelian extensions.

Asymptotic formulas expressing prime bias magnitudes.

Results hold unconditionally in positive characteristic cases.

Abstract

Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for achieving 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, the authors succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which provides a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $\sigma$ in the Galois group, we have established a criterion of the bias of primes whose Frobenius elements are equal to $\sigma$ under the assumption of DRH. As an application we have obtained a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is known, and all these results hold unconditionally.