Derivatives of Dedekind sums and their reciprocity law

In this paper derivatives of Dedekind sums are defined, and their reciprocity laws are proved. They are obtained from values at non-positive integers of the first derivatives of Barnes’ double zeta functions. As special cases, they give finite product expressions of the Stirling modular form and the...

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Bibliographic Details
Published in:Journal of number theory 2003-02, Vol.98 (2), p.280-309
Main Author: Ota, Kaori
Format: Article
Language:English
Subjects:
Barnes’ multiple zeta function
Derivatives of Dedekind sums
Double gamma function
Reciprocity law
Stirling modular form
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Summary:In this paper derivatives of Dedekind sums are defined, and their reciprocity laws are proved. They are obtained from values at non-positive integers of the first derivatives of Barnes’ double zeta functions. As special cases, they give finite product expressions of the Stirling modular form and the double gamma function at positive rational numbers.
ISSN:0022-314X
1096-1658
DOI:10.1016/S0022-314X(02)00046-X