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The zeta-functions are very interesting objects in analytic number theory. In the paper, the main attention is focused to the Riemann zeta-function, from the first results on the denseness until limit theorems with application in the investigation of universality in Voronin's sense. A special attention is devoted to the Matsumoto zeta-function. The paper is based on the content of a lecture to the master and doctoral students at the Graduate School of Mathematics of Nagoya University (Japan). [ABSTRACT FROM AUTHOR]

We study two functions ϵk(s) and ϵk1,…,kr(s) introduced by Arakawa and Kaneko [Nagoya Math. J., 153 (1999), 189{209] and relate them with (finite) multiple zeta values and multiple zeta star values using elementary methods. In particular, we give an alternative proof of a result of Ohno [Y. Ohno: J. Number Theory, 74 (1999), 39-43]. [ABSTRACT FROM AUTHOR]

It has been pointed out by Li, Hashimoto and Kanemitsu [Ito, T., 2006, On an integral representation of special values of the zeta function at odd integers. Journal of the Mathematical Society of Japan, 58, 681-691.] that the recent result of Ito [Li, H.-L., Hashimoto, M. and Kanemitsu, S., 2007, Examples of the Hurwitz transforms, in press.] is example 5.2 of Espinosa and Moll [Mikolás, M., 1957, A simple proof of the functional equation for the Riemann zeta-function and a forumula of Hurwitz. Acta Scientiarum Mathematicarum (Szeged), 18, 261-263.], whose main results in turn are contained in a more anteceding deep result of Mikolás [Nakamura, T., 2006, A functional relation for the Tornheim double zeta function. Acta Arithmetica, 125, 257-263.]. In this paper, we shall clarify the situation more completely and give some further results, from the point of view of application of the functional equation. [ABSTRACT FROM AUTHOR]