Related papers: On certain combination of colored multizeta values

In this work we study integrals of the form $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{L}\exp(-Tx)dx$. For $L \in \mathbb{N}$, $L \leq 4$ and $T \in \mathbb{R}$ we give explicit expressions in terms of derivatives of the Hurwitz…

We give a parameterized generalization of the sum formula for quadruple zeta values. The generalization has four parameters, and is invariant under a cyclic group of order four. By substituting special values for the parameters, we also…

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $S_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer $n$ such that every coloring $\chi:[1,n] \rightarrow \{0,1,\dots,r-1\}$ admits a solution to $\sum_{i=1}^{k-1} x_i = x_k$…

We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…

In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \bar{z_j}. $…

Let $$ A_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j\choose 2j}\sum_{k=0}^{n-1} \sin(a+2k\pi/n) \cos^{2j}(a+2k\pi/n) $$ and $$ B_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j+1\choose 2j+1}\sum_{k=0}^{n-1} \sin(a+2k\pi/n) \cos^{2j+1}(a+2k\pi/n), $$ where $m\geq…

In this note we give the most elementary method (as far as we know) to express $\zeta(2n+1)$ in terms of $\{\zeta(2k)|k\geq 1\}$. The method is based on only some elementary works by Leonhard Euler, so it is very instructive to non-experts…

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite…

Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant,…

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently…

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals…

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}%…

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating…

Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by \[\zeta_F(s) \,:=\, \sum_{n=1}^{\infty} \frac{1}{F_n^s}\,,\quad \zeta_F^*(s) \,:=\,\sum_{n=1}^{\infty}…

For $k\leq n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\geq 2$. More precisely, $E(mn,k)=\sum_{|\boldsymbol{\alpha}|=n}\zeta(m\alpha_1,m\alpha_2,\ldots,…

We present a method to study the behavior of a power series of type $$f(x):=\sum_{n=0}^\infty (-1)^n c_n\frac{x^{2n+1}}{(2n+1)!}$$ when $x\to\infty$. We apply our method to study the function…

We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers

The relation between Eisenstein series and the $J$-homomorphism is an important topic in chromatic homotopy theory at height $1$. Both sides are related to the special values of the Riemann $\zeta$-function. Number theorists have studied…

In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to…

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…