18 research outputs found
The evaluation of Tornheim double sums. Part 1
We provide an explicit formula for the Tornheim double series in terms of
integrals involving the Hurwitz zeta function. We also study the limit when the
parameters of the Tornheim sum become natural numbers, and show that in that
case it can be expressed in terms of definite integrals of triple products of
Bernoulli polynomials and the Bernoulli function .Comment: 23 pages, AMS-LaTex, to appear in Journal of Number Theor
On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function
We derive new results about properties of the Witten zeta function associated
with the group SU(3), and use them to prove an asymptotic formula for the
number of n-dimensional representations of SU(3) counted up to equivalence. Our
analysis also relates the Witten zeta function of SU(3) to a summation identity
for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new
proof of that identity and show that it is a special case of a stronger
identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic
Functional relations for zeta-functions of weight lattices of Lie groups of type
We study zeta-functions of weight lattices of compact connected semisimple
Lie groups of type . 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.Comment: 25 Page
A study on multiple zeta values from the viewpoint of zeta-functions of root systems
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 -ple zeta-functions of Euler-Zagier type can be regarded as the
zeta-function associated with a certain sub-root system of type . 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
. 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
which were previously known only in the cases of depth .
Furthermore, considering a sub-root system of type analogously, we can
give relevant analogues of the Hoffman-Zagier formula, parity results and
restricted sum formulas.Comment: 27 page
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Identities in the spirit of Euler
In this paper we develop new identities in the spirit of Euler. We shall investigate and report on new Euler identities of weight p+2, for p an odd integer, but with a non unitary argument of the harmonic numbers. Some examples of these Euler identities will be given in terms of Riemann zeta values, Dirichlet values and other special functions.peerReviewe