Hirofumi Tsumura

On functional relations between the Mordell–Tornheim double zeta functions and the Riemann zeta function

In this paper, we give certain analytic functional relations between the Mordell?Tornheim double zeta functions and the Riemann zeta function. These can be regarded as continuous generalizations of the known discrete relations between the Mordell?Tornheim double zeta values and the Riemann zeta values at positive integers discovered in the 1950s.

A note on q-analogues of the Dirichlet series and q-Bernoulli numbers

Abstract We shall construct q -analogues of the Dirichlet series which relate to algebraic number fields. By way of examples, we treat q -analogues of the Dirichlet L -series. As applications, we obtain the q -representations for the class number formulas.

On alternating analogues of Tornheim's double series

In this paper, we evaluate the alternating analogues of Tornheim’s double series. This is an answer to the problem posed by Subbarao-Sitaramachandrarao, and can be regarded as an alternating analogue of the evaluation formula for Tornheim’s double series, given by Huard, Williams and Z. Nan-Yue. We also evaluate partial Tornheim’s double series.

Evaluation formulas for Tornheim's type of alternating double series

In this paper, we give some evaluation formulas for Tornheims type of alternating series by an elementary and combinatorial calculation of the uniformly convergent series. Indeed, we list several formulas for them by means of Riemanns zeta values at positive integers.

An Elementary Proof of Euler's Formula for ζ(2m)

Various proofs of (1) have been obtained (see, for example, [3]). In an issue of this MONTHLY [5], Williams gave an elementary proof of (1). In [1], Boo Rim Choe produced another elementary proof of (1) in the case m = 1. In this note, we present a new elementary proof of (1). The author has introduced a method to evaluate ? (2m) by the calculation of q-series (see [4]). The method here is more simple and direct. Ford > 0 and u in [1, 1 +d], we let

On Mordell-Tornheim zeta values

We prove that the Mordell-Tornheim zeta value of depth r can be expressed as a rational linear combination of products of the Mordell-Tornheim zeta values of lower depth than r when r and its weight are of different parity.

ZETA-FUNCTIONS OF ROOT SYSTEMS

In this paper, we introduce multi-variable zeta-functions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zeta-functions associated with Lie algebras which can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1, 2, 3) which include what is called Witten’s volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups.

Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems

The shuffle product plays an important role in the study of multiple zeta values (MZVs). This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of MZVs of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel’d integral expressions for MZVs. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.

ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS IV

We study the values of the zeta-function of the root system of type G2 at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases including odd integers is also discussed.

Certain functional relations for the double harmonic series related to the double Euler numbers

In this paper, we give certain analytic functional relations for the double harmonic series related to the double Euler numbers. These can be regarded as continuous generalizations of the known discrete relations obtained by the author recently.