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Asymptotic properties of zeta functions over finite fields

Abstract : In this paper we study asymptotic properties of families of zeta and $L$-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer--Siegel type results and the results on distribution of zeroes. We generalize to this abstract setting the results of Tsfasman, Vl\u adu\c t and Lachaud, who studied similar problems for curves and (in some cases) for varieties over finite fields. In the classical case of zeta functions of curves we extend a result of Ihara on the limit behaviour of the Euler--Kronecker constant. Our results also apply to $L$-functions of elliptic surfaces over finite fields, where we approach the Brauer--Siegel type conjectures recently made by Kunyavskii, Tsfasman and Hindry.
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Submitted on : Saturday, January 20, 2018 - 2:32:13 AM
Last modification on : Monday, November 15, 2021 - 7:30:02 PM

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Alexey Zykin. Asymptotic properties of zeta functions over finite fields. Finite Fields and Their Applications, 2015, 35, pp.247-283. ⟨10.1016/j.ffa.2015.04.005⟩. ⟨hal-01689003⟩



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