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We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…
In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the…
By introducing a novel integration kernel for Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and…
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…
In 1999, Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers and investigated its relation to multiple zeta values. Since the poly-Bernoulli numbers appear in this…
Inversion of function sinc(x) is studied. New series and integral representations of branches of inverse function are obtained using Fourier analysis.
We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…
We introduce a new method which enables us to calculate the coefficients of the poles of local zeta functions very precisely and prove some explicit formulas. Some vanishing theorems for the candidate poles of local zeta functions will be…
The table of Gradshteyn and Rhyzik contains some trigonometric integrals that can be expressed in terms of the beta function. We describe the evaluation of some of them.
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…
We provide a new analysis of the irreducible loop integrals first considered in a 2003 paper of Wu. Using convergence ideas from probability, we produce conditions on the regulator masses so that the integrals have well-defined limits in…
The local zeta functions (also called Igusa's zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact…
We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by…
We construct 3 finite systems of $4-F-3$ hypergeometric orthogonal polynomials. The weights are 1) the weight defined by the $5-H-5$ Dougall summation formula; 2) the integrand in the Askey beta-integral; 3) the weight $w(s)=|p(s)/q(s)|^2$,…
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…
I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations.
By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset $S\subset \mathbb{P}^1(\mathbb{C})$ and studying their transformation properties under rational functions, we show that multiple…
We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The…