Research on multiple zeta functions and development of the computer programs about evaluation of multiple zeta values

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The research representative accomplished theoretical research, exchanged information by having promoted the exchange with a researcher in and outside the country, arranged the result of multiple zeta-functions obtained based on them now As a concrete result, about the functional relations for the multiple zeta-functions, the research representative accomplished the joint research with Professor Kohji Matsumoto (Nagoya University), based on Hardy's work about Riemann zeta-function. The concrete functional relation formulas were given by a method which becomes similar to the proof of the functional equation of it Moreover the research on the Witten zeta-functions associated with the semi-simple Lie algebras were developed from the multi-variable function theory, which is a joint work with Professors Yasushi Komori (Nagoya University)and Kohji Matsumoto. The multiple zeta-functions of abstract root systems could be constituted, and the analytic continuation and those functional relations for them were obtained. As special value relations, the concrete formulas of the special values of the Witten zeta-function were obtained. Furthermore, the relations for multiple polylogarithms were given by the same method. Further the research representative studied multiple version of Dirichlet series involving hyperbolic functions, first studied by Cauchy and Ramanujan, and also studied the Eisenstein type of multiple series. The calculation program about the value and expression of relations of a multiple zeta-function was developed based on those theories. They will be opened on the website of the research representative in the near future.