  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.
