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Abstract

Abstract. 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.