Chebyshev's bias for Fermat curves of prime degree

TL;DR

This paper links the distribution of primes related to Fermat curves of prime degree to the Deep Riemann Hypothesis, showing an equivalence and computing zero orders of associated L-functions.

Contribution

It establishes an equivalence between prime number race asymptotics for Fermat curves and the Deep Riemann Hypothesis, and computes zero orders of certain L-functions.

Findings

Asymptotic prime distribution for Fermat curves is equivalent to DRH.

Equivalence extends to some quotients of Fermat curves.

Zero order at s=1 for second moment L-functions computed under DRH.

Abstract

In this article, we prove that an asymptotic formula for the prime number race with respect to Fermat curves of prime degree is equivalent to part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of partial Euler products of $L$-functions on the critical line. We also show that such an equivalence holds for some quotients of Fermat curves. As an application, we compute the order of zero at $s=1$ for the second moment $L$-functions of those curves under DRH.