Zeta Mahler measures, multiple zeta values and L-values

2015 ◽  
Vol 11 (07) ◽  
pp. 2239-2246
Author(s):  
Yoshitaka Sasaki
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Mahler Measure ◽  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Multiple Zeta ◽  
Zeta Values

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

scholarly journals Some relations of interpolated multiple zeta values

2017 ◽  
Vol 28 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Zhonghua Li ◽  
Chen Qin
Keyword(s):  
Generating Function ◽  
Hypergeometric Functions ◽  
Multiple Zeta Values ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values

In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.

scholarly journals Algorithms for Some Euler-Type Identities for Multiple Zeta Values

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shifeng Ding ◽  
Weijun Liu
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
Convergent Series ◽  
Multiple Zeta Values ◽  
Double Product ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

Multiple zeta values are the numbers defined by the convergent seriesζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), wheres1,s2,…,skare positive integers withs1>1. Fork≤n, letE(2n,k)be the sum of all multiple zeta values with even arguments whose weight is2nand whose depth isk. The well-known resultE(2n,2)=3ζ(2n)/4was extended toE(2n,3)andE(2n,4)by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbersE(2n,k)and then gave a direct formula forE(2n,k)for arbitraryk≤n. In this paper we apply a technique introduced by Granville to present an algorithm to calculateE(2n,k)and prove that the direct formula can also be deduced from Eisenstein's double product.

scholarly journals On multiple zeta values of even arguments

2017 ◽  
Vol 13 (03) ◽  
pp. 705-716 ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Bernoulli Number ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta

For [Formula: see text], let [Formula: see text] be the sum of all multiple zeta values with even arguments whose weight is [Formula: see text] and whose depth is [Formula: see text]. Of course [Formula: see text] is the value [Formula: see text] of the Riemann zeta function at [Formula: see text], and it is well known that [Formula: see text]. Recently Shen and Cai gave formulas for [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. We give two formulas for [Formula: see text], both valid for arbitrary [Formula: see text], one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers [Formula: see text] and for the analogous numbers [Formula: see text] defined using multiple zeta-star values of even arguments.

scholarly journals Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

2021 ◽  
Vol 7 (1) ◽  
pp. 288-293
Author(s):  
Junjie Quan ◽  
Keyword(s):  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Recurrence Formulas ◽  
Multiple Zeta ◽  
Zeta Values

<abstract><p>In a recent paper <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>, Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.</p></abstract>

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