Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Abdelmejid Bayad; Yilmaz Simsek

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A generalization of the reciprocity law of multiple Dedekind sums

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Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad. In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws. ...

On Witten multiple zeta-functions associated with semisimple Lie algebras I

Kohji Matsumoto, Hirofumi Tsumura (2006)

Annales de l’institut Fourier

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We define Witten multiple zeta-functions associated with semisimple Lie algebras $𝔰𝔩 (n)$ , $(n = 2, 3, ...)$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case $𝔰𝔩 (4)$ , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations,...

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