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Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle…

We prove three theorems on finite real multiple zeta values: the symmetric formula, the sum formula and the height-one duality theorem. These are analogues of their counterparts on finite multiple zeta values.

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded $k$-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several…

We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta(2p)$, where $p$ is a positive integer, and an expression of $\zeta(2p+1)$ involving…

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The…

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those…

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of…

Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Pad\'e approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi.…

This thesis is a study of algebraic and geometric relations between multizeta values. In chapter 2, we prove a result which gives the dimension of the associated depth-graded pieces of the double shuffle Lie algebra in depths 1 and 2. In…

Interpolated multiple $q$-zeta values are deformation of multiple $q$-zeta values with one parameter, $t$, and restore classical multiple zeta values as $t = 0$ and $q \to 1$. In this paper, we discuss generating functions for sum of…

Let s_1,...,s_d be d positive integers and consider the multiple Hurwitz-zeta value zeta(s_1,...,s_d;-1/2,...,-1/2)/2^w where w=s_1+...+s_d is called the weight. For d<n+1, let T(2n,d) be the sum of all these values with even arguments…

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit…

For differential operators of Calabi-Yau type, Candelas, de la Ossa and van Straten conjecture the appearance of $p$-adic zeta values in the matrix entries of their $p$-adic Frobenius structure expressed in the standard basis of solutions…

A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers…

We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for…

In correspondence with Goldbach, Euler began investigating series of the form $\sum_{k \geq 1} k^{-m}\left(1 + 2^{-n} + \cdots + k^{-n}\right)$, which are known today as Euler sums. For the case where $n=1$ and $m \geq 2$, Euler was able to…

When looking at how periods of $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, 1, \infty \rbrace )$, i.e. multiple zeta values, embeds into periods of $\pi_{1}^{\mathfrak{m}}(\mathbb{P}^{1}\diagdown \lbrace 0, \pm 1, \infty…

We generalize the well-known parity theorem for multiple zeta values (MZV) to functional equations of multiple polylogarithms (MPL). This reproves the parity theorem for MZV with an additional integrality statement, and also provides parity…

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta…

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall's identity for the generalized hypergeometric function…