Related papers: Unit cyclotomic multiple zeta values for $\mu_2,\m…

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain…

In this paper, we show that the cyclotomic symmetric multiple zeta values, independently proposed by Jarossay, Singar and Zhao, and Tasaka, span the space of the cyclotomic multiple zeta values modulo $\pi i$.

Let $r\ge 1$ be an integer. For any multiple index $\mathbf{s}=(s_1,s_2,\cdots,s_r) \in\mathbb{Z}_{\geq 1}^r$ with $s_r>1$, the multiple zeta value (MZV for short) is defined by \begin{align*} \zeta(s_1,s_2,\cdots,s_r):=\sum_{1\leq…

For any positive integer $N$ let $\mu_N$ be the group of the $N$th roots of unity. In this note we shall study the $\Q$-linear relations among values of multiple polylogarithms evaluated at $\mmu_N$. We show that the standard relations…

We establish a new class of relations among the multiple zeta values \zeta(k_1,k_2,...,k_n), which we call the cyclic sum identities. These identities have an elementary proof, and imply the "sum theorem" for multiple zeta values. They also…

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime…

The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta…

The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin…

Let $n$ be an even positive integer with at most three distinct prime factors and let $\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \in \Q[x]$ having at least…

This is a review on the two first parts of our work on $p$-adic multiple zeta values at $N$-th roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} -…

Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,\zeta_n)$, where $\zeta_n$ is a primitive…

Let Q(4n,d) be the sum of all multiple zeta values of depth d and weight 4n whose arguments are all multiples of 4. In this paper we derive a formula of Q(4n,d) for all d>2 as a finite sum involving binomial coefficients, Bernoulli numbers…

For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the…

We study the modular graph functions introduced by Green, Russo, Vanhove in the context of type II superstring scattering amplitudes of 4 gravitons on a torus. In particular we describe a method to algorithmically compute the coefficients…

In this paper we shall consider a few variants of the motivic multiple zeta values of level two by restricting the summation indices in the definition of multiple zeta values to some fixed parity patterns. These include Hoffman's multiple…

Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit…

Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen.

In this paper we consider iterated integrals of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple…

We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…

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…