Chebyshev's bias for Fermat curves of prime degree

Yoshiaki Okumura

Published 2023-07-12Version 1

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