A functional relation for Tornheim's double zeta functions

Kazuhiro Onodera¹

¹ Division of Mathematics Chiba Institute of Technology Shibazono 2-1-1, Narashino Chiba 275-0023, Japan

Acta Arithmetica, Tome 162 (2014) no. 4, pp. 337-354

Résumé

We generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results on the behavior of a certain Witten's zeta function at each integer. As an appendix, we prove a functional equation for Euler's double zeta function.

Détail

DOI : 10.4064/aa162-4-2

Keywords: generalize partial fraction decomposition which fundamental theory multiple zeta values prove relation between tornheims double zeta functions three complex variables applications integral representations several zeta functions extension parity result whole domain convergence concrete expressions tornheims double zeta function non positive integers results behavior certain wittens zeta function each integer appendix prove functional equation eulers double zeta function