Generalised root identities for zeta functions of curves over finite fields

2012/02/12 by Richard Stone, Stone, Richard
Computer Science · Mathematics · #11M26 (Primary) 11M36 #40G05 (Secondary) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #Coding theory and cryptography #Complex Variables (math.CV) #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1202.4351

openalex publication_date 2012/02/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider generalised root identities for zeta functions of curves over finite fields, ζ_k, and compare with the corresponding analysis for the Riemann zeta function. We verify numerically that, as for ζ, the ζ_k do satisfy the generalised root identities and we investigate these in detail for the special cases of μ=0,-1 & -2. Unlike for ζ, however, we show that in the setting of zeta functions of curves over finite fields the μ=-2 root identity is consistent with the Riemann hypothesis (RH) proved by Weil. Comparison of this analysis with the corresponding calculations for ζilluminates the fact that, even though both ζand ζ_k have both Euler and Hadamard product representations, it is the detailed structure of the counting function, N(T), which drives the Cesaro computations on the root side of these identities and thereby determines the implications of the root identities for RH in each setting.

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