A Wasserstein metric approach to generalized Skewes' numbers. I. Prime number races

TL;DR

This paper investigates generalized Skewes' numbers in prime number races, constructing sequences with rapidly growing Skewes' numbers and establishing bounds under certain hypotheses using Wasserstein metrics.

Contribution

It introduces a Wasserstein metric approach to analyze prime number races, disproves a conjecture, and provides conditional bounds assuming RH and linear independence.

Findings

Constructed sequences with rapidly growing Skewes' numbers.

Disproved Fiorilli's conjecture unconditionally.

Established conditional upper bounds under RH and linear independence.

Abstract

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences of highly composite moduli $q$ such that those Skewes' numbers grow very rapidly in some sense. This disproves unconditionally a conjecture of Fiorilli. In the other direction, assuming the Generalized Riemann Hypothesis and an effective linear independence hypothesis, we establish conditional upper bounds for generalized Skewes' numbers. Our approach relies on a quantitative Kronecker-Weyl theorem formulated in terms of the $1$-Wasserstein metric to obtain explicit rates for the convergence to the limiting distributions in these races.