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On a generalization of Hamburger’s theorem

Published online by Cambridge University Press:  22 January 2016

Akinori Yoshimoto
Akinori Yoshimoto*
Affiliation:
Department of Mathematics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan
*
Current Address: Department of Mathematics, Faculty of Science Nagoya University, Chikusa-ku, Nagoya 464, Japan
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The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grössencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grössencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grössencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 98 , June 1985 , pp. 67 - 76
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

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