A character on the quasi-symmetric functions coming from multiple zeta values.
A formula for the logarithm of the KZ associator.
A functional relation for Tornheim's double zeta functions
We generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results on the behavior of a certain Witten's zeta function...
Algebraic setup of non-strict multiple zeta values
Asymptotic analysis and special values of generalised multiple zeta functions
This is an expository article, based on the talk with the same title, given at the 2011 FASDE II Conference in Będlewo, Poland. In the introduction we define Multiple Zeta Values and certain historical remarks are given. Then we present several results on Multiple Zeta Values and, in particular, we introduce certain meromorphic differential equations associated to their generating function. Finally, we make some conclusive remarks on generalisations of Multiple Zeta Values as well as the meromorphic...
Asymptotic representations for Fibonacci reciprocal sums and Euler’s formulas for zeta values
We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.
Combinatorial aspects of multiple zeta values.
Cyclic sum of multiple zeta values
Evaluation of triple Euler sums.
Evaluations of -fold Euler/Zagier sums: a compendium of results for arbitrary .
Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial
Given a multivariate polynomial with integral coefficients verifying an hypothesis of analytic regularity (and satisfying ), we determine the maximal domain of meromorphy of the Euler product and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.
Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents
Generalizations of Apostol-Vu and Mordell-Tornheim multiple zeta functions
Higher Mahler measures and zeta functions
Linear differential equations and multiple zeta values. I. Zeta(2)
Certain generating fuctions for multiple zeta values are expressed as values at some point of solutions of linear meromorphic differential equations. We apply asymptotic expansion methods (like the WKB method and the Stokes operators) to solutions of these equations. In this way we give a new proof of the Euler formula ζ(2) = π²/6. In further papers we plan to apply this method to study some third order hypergeometric equation related to ζ(3).
Mod structure of alternating and non-alternating multiple harmonic sums
The well-known Wolstenholme’s Theorem says that for every prime the -st partial sum of the harmonic series is congruent to modulo . If one replaces the harmonic series by for even, then the modulus has to be changed from to just . One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction...
Multiple zeta values and periods of moduli spaces
We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces of Riemann spheres with marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle...
Multiple zeta values for coordinatewise limits at non-positive integers
On a reciprocity law for finite multiple zeta values.
On multiple analogues of Ramanujan’s formulas for certain Dirichlet series
In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.