Prime Ideal Races With Several Competitors

TL;DR

This paper extends the analysis of prime ideal races in number fields to multiple competitors, providing explicit formulas and criteria for moderate biases, and exploring their behaviors and densities.

Contribution

It generalizes previous two-way race results to r-way races, offering explicit bias formulas and criteria for moderate biases in number fields.

Findings

Explicit bias formula for r-way prime ideal races

Criterion for r-moderate races in abelian extensions

Density results and behavioral differences for r=3 and r≥4

Abstract

We investigate races among prime ideals in number fields when there are two or more competing conjugacy classes. In their work [4], Fiorilli and Jouve studied two-way races in number fields and showed that-unlike the classical setting of primes in arithmetic progressions-these biases can approach the extreme values of 0 and 1. They also identified when these biases tend toward one-half (as the degree of the extension grows), which we call ''moderate biases'' because that behavior mirrors the classical case. In this paper, we extend their analysis to races with r competing conjugacy classes (rway races) and precisely study the cases where these biases are moderate (meaning they tend to 1/r! as the discriminant of the extension grows). Our first main result is an explicit formula for the bias in any r-way race (for all r $\ge$ 2), generalizing the two-way formula of Fiorilli and Jouve [4] and Lamzouri's r-way expression in the classical case of residue classes modulo q [12], under the same hypotheses. Using this formula, we give a criterion characterizing completely, in the abelian case, r-moderate races. Surprisingly, once r $\ge$ 3 this criterion is independent of r, making the two-way race exceptional. We also construct families of number fields exhibiting such moderacy, we prove density results for the values of logarithmic densities and exhibit different behaviors of those densities between the cases r = 3 and r $\ge$ 4.