Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values

10.1142/9634 ◽  
2015 ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Zeta Functions ◽  
Special Values ◽  
Multiple Zeta ◽  
Multiple Polylogarithms

scholarly journals Variations of Mixed Hodge Structures of Multiple Polylogarithms

2004 ◽  
Vol 56 (6) ◽  
pp. 1308-1338 ◽  
Author(s):  
Jianqiang Zhao
Keyword(s):  
Number Field ◽  
Zeta Functions ◽  
Explicit Construction ◽  
Hodge Structures ◽  
Special Values ◽  
Multiple Polylogarithms ◽  
Real Analytic ◽  
Mixed Hodge Structures

AbstractIt is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

Generalized harmonic numbers and Euler sums

2017 ◽  
Vol 13 (02) ◽  
pp. 513-528 ◽  
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Zeta Functions ◽  
Multiple Zeta Values ◽  
Harmonic Numbers ◽  
Special Values ◽  
Euler Sums ◽  
Summation Formulas ◽  
Multiple Zeta ◽  
Zeta Values ◽  
New Formula ◽  
Generalized Harmonic Numbers

In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa–Kaneno zeta functions and a new formula of multiple zeta values of height one as examples.

scholarly journals Multiple zeta values, poly-Bernoulli numbers, and related zeta functions

1999 ◽  
Vol 153 ◽  
pp. 189-209 ◽  
Author(s):  
Tsuneo Arakawa ◽  
Masanobu Kaneko
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Multiple Zeta Values ◽  
Special Values ◽  
Multiple Zeta ◽  
Zeta Values

AbstractWe study the function and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.

scholarly journals Values of multiple zeta functions with polynomial denominators at non-positive integers

2021 ◽  
pp. 2150038
Author(s):  
Driss Essouabri ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Zeta Functions ◽  
Summation Formula ◽  
Bernoulli Numbers ◽  
Explicit Formulas ◽  
Integer Points ◽  
Special Values ◽  
Multiple Zeta ◽  
Positive Integers

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 when the polynomials are power sums, and observe that some “trivial zeros” exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler’s theorem, and a Raabe-type lemma due to Friedman and Pereira.

Periods, Multiple Zeta Functions and ζ(3)

2014 ◽  
pp. 185-203
Author(s):  
M. Ram Murty ◽  
Purusottam Rath
Keyword(s):  
Zeta Functions ◽  
Multiple Zeta
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