Explicit Kronecker-Weyl theorems and applications to prime number races

TL;DR

This paper develops explicit versions of Kronecker-Weyl theorems without linear independence assumptions and applies them to prime number races, providing new results on densities over number and function fields.

Contribution

It introduces explicit Kronecker-Weyl theorems without linear independence assumptions and applies them to prime number races, offering novel insights into their asymptotic densities.

Findings

Proved explicit Kronecker-Weyl theorems in discrete and continuous settings.

Established new results on the existence and positivity of prime number race densities.

Extended results to races over function fields without linear independence hypotheses.

Abstract

We prove explicit versions of the Kronecker-Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis.