Chebyshev's bias for Fermat curves of prime degree

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.