Chebyshev's bias for Fermat curves of prime degree

2023/07/12 by Yoshiaki Okumura, Okumura, Yoshiaki · 1 citation
Arts and Humanities · Mathematics · #11N05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT) #Primary 11M06 #Secondary 11G30

paper · pdf · doi:10.48550/arxiv.2307.05958

openalex publication_date 2023/07/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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