arXiv subjects
Publications and source records attributed to Hirofumi Tsumura.
In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical results given by Cauchy, Mellin and Kronecker. In the latter part, we give new formulas for some infinite series involving the hyperbolic cosine function.
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running over positive integers, which may be regarded as zeta-functions associated with hyperplane arrangements.
We study the values of the zeta-function of the root system of type $G_2$ 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.
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.
We study zeta-functions of weight lattices of compact connected semisimple Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6) and PU(4), and give some functional relations and new classes of evaluation formulas for them.
We prove sum formulas for double polylogarithms of Hurwitz type, that is, involving a shifting parameter $b$ in the denominator. These formulas especially imply well-known sum formulas for double zeta values, and sum formulas for double $L$-values. Further, differentiating in $b$, we obtain a kind of weighted sum formula for double polylogarithms and double $L$-values. We also give sum formulas for partial double zeta values with some congruence conditions. Our proofs of those sum formulas are based on certain functional relations for double polylogarithms of Hurwitz type.
We define zeta-functions of weight lattices of compact connected semisimple Lie groups. If the group is simply-connected, these zeta-functions coincide with ordinary zeta-functions of root systems of associated Lie algebras. In this paper we consider the general connected (but not necessarily simply-connected) case, prove the explicit form of Witten's volume formulas for these zeta-functions, and further prove functional relations among them which include their volume formulas.
In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from functional equations for Barnes zeta-functions. In the latter half part, we generalize some evaluation formulas of certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and formulated by Berndt. From our consideration, we give multiple versions of these known formulas.
We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for $q$-zeta functions at positive integers.
The shuffle product plays an important role in the study of multiple zeta values. 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 multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel'd integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.
We prove certain general forms of functional relations among Witten multiple zeta-functions in several variables (or zeta-functions of root systems). The structural background of those functional relations is given by the symmetry with respect to Weyl groups. From those relations we can deduce explicit expressions of values of Witten zeta-functions at positive even integers, which is written in terms of generalized Bernoulli numbers of root systems. Furthermore we introduce generating functions of those Bernoulli numbers of root systems, by which we can give an algorithm of calculating Bernoulli numbers of root systems.
In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.
In this paper, we introduce a certain method to construct polynomials producing many absolute pseudoprimes. By this method, we give new polynomials producing absolute pseudoprimes with any fixed number of prime factors which can be viewed as a generalization of Chernick's result. By the similar method, we give another type of polynomials producing many absolute pseudoprimes. As concrete examples, we tabulate the counts of such numbers of our forms.