### A character on the quasi-symmetric functions coming from multiple zeta values.

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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...

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...

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.

Given a multivariate polynomial $h\left({X}_{1},\cdots ,{X}_{n}\right)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h\left(\mathbf{0}\right)=1$), we determine the maximal domain of meromorphy of the Euler product ${\prod}_{p\phantom{\rule{4pt}{0ex}}\mathrm{prime}}h\left({p}^{-{s}_{1}},\cdots ,{p}^{-{s}_{n}}\right)$ 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.

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).

The well-known Wolstenholme’s Theorem says that for every prime $p\>3$ the $(p\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1)$-st partial sum of the harmonic series is congruent to $0$ modulo ${p}^{2}$. If one replaces the harmonic series by ${\sum}_{k\ge 1}1/{n}^{k}$ for $k$ even, then the modulus has to be changed from ${p}^{2}$ to just $p$. 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...

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces ${\U0001d510}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on ${\U0001d510}_{0,n}$ 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...

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.