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All but Finitely Many Non‐Trivial Zeros of the Approximations of the Epstein Zeta Function are Simple and on the Critical Line
Author(s) -
Ki Haseo
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504015060
Subject(s) - mathematics , critical line , riemann zeta function , simple (philosophy) , function (biology) , functional equation , series (stratigraphy) , line (geometry) , pure mathematics , euler's formula , arithmetic zeta function , combinatorics , mathematical analysis , partial differential equation , paleontology , philosophy , physics , geometry , epistemology , evolutionary biology , biology , thermodynamics
The Chowla–Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla–Selberg formula, and although these partial sums satisfy a functional equation, as does an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of the function. We show that in any strip containing the critical line, all but finitely many zeros of the function are simple and on the critical line. For many Epstein zeta functions we show that all but finitely many non‐trivial zeros of partial sums in the Chowla–Selberg formula are simple and on the critical line. 2000 Mathematics Subject Classification 11M26.