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Notes on the Poisson formula. With an appendix by Irina S. Rezvyakova. (English. Russian original) Zbl 1323.11065

St. Petersbg. Math. J. 23, No. 5, 781-818 (2012); translation from Algebra Anal. 23, No. 5, 1-54 (2011).
Summary: This is a survey of applications of harmonic analysis to the study of the zeta-functions of one-dimensional schemes. A new version of the Tate-Iwasawa method is suggested that involves holomorphic duality for discrete groups instead of Pontryagin duality. A relationship is found between the Poisson formula and the residue formula on the compactification of the holomorphically dual group. Links to explicit formulas for zeta-functions of algebraic curves are found. A numerical analog of these constructions is considered in the appendix written by I. S. Rezvyakova.

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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