Related papers: A Weak Euler Formula for l-adic Galois Double Zeta…
We introduce an algebraic formulation of cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by…
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of…
For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…
In this paper, we obtain a general t-shuffle product formula, using which we derive a generalized Euler decomposition formula for interpolated multiple zeta values. We also provide the same formula in case of height one through two…
We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic…
We compute the weighted Euler characteristic, equivariant with respect to the action of the symplectic group of degree six over the field of two elements, of the moduli space of principally polarized abelian threefolds together with a level…
We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``regularized value of $\zeta(1)$'' and of Mascheroni's and…
A formula for the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known…
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…
We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…
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.
In this paper, we will study finite multiple $T$-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for…
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an…
We study the values of the zeta-function of the root system of type $G_2$ at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include…
In this paper, we determine the order of magnitude of the $2q$-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0<q\le 1/2$ and $0< \alpha<1$, completing the results of Bondarenko, Heap and Seip, and of…
In this paper, we employ methods of contour integration and residue calculus to investigate the parity of two classes of cyclotomic Euler-type sums. One class involves products of cyclotomic harmonic numbers, while the other involves…
Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type…
We establish a connection between a function and a series representation using a similar technique with that that Euler used to solve the Basel problem. Our result concerns a more general series from which one can obtain $\zeta(2k)$ as a…
We present here an approach to a computation of $\zeta(2)$ by changing variables in the double integral using hyperbolic trig functions. We also apply this approach to present $\zeta(n)$, when $n>2$, as a definite improper integral of…
We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} \zeta(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty}…