## Rocky Mountain Journal of Mathematics

### A decomposition of $\zeta(2n+3)$ into sums of multiple zeta values

#### Abstract

We evaluate the multiple zeta value $ζ ( 1 , { 2 } n + 1 )$ or its dual $ζ ( { 2 } n , 3 )$. When $n$ is even, along with stuffle relations already available, it is enough to evaluate all multiple zeta values of the form $ζ ( { 2 } a , 3 , { 2 } b )$ with $a + b = n$. Furthermore, we obtain a decomposition for $2 ζ ( 2 n + 3 )$ as

$ζ ⋆ ( 3 , { 2 } n ) + ζ ( { 2 } n , 3 ) + ∑ r = 1 n ∑ | α | = n + 1 ζ ( 1 , 2 α 1 , 2 α 2 , … , 2 α r ) ,$

which also can be used to evaluate $ζ ( { 2 } n , 3 )$ when $n$ is even.

#### Article information

Source
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 551-558.

Dates
Received: 12 June 2019
Revised: 3 November 2019
Accepted: 4 November 2019
First available in Project Euclid: 29 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1590739289

Digital Object Identifier
doi:10.1216/rmj.2020.50.551

Mathematical Reviews number (MathSciNet)
MR4104393

Zentralblatt MATH identifier
07210978

#### Citation

Eie, Minking; Liaw, Wen-Chin; Ong, Yao Lin. A decomposition of $\zeta(2n+3)$ into sums of multiple zeta values. Rocky Mountain J. Math. 50 (2020), no. 2, 551--558. doi:10.1216/rmj.2020.50.551. https://projecteuclid.org/euclid.rmjm/1590739289

#### References

• F. Brown, “Mixed Tate motives over $\mathbb{Z}$”, Ann. of Math. $(2)$ 175:2 (2012), 949–976.
• K.-W. Chen, “Generalized harmonic numbers and Euler sums”, Int. J. Number Theory 13:2 (2017), 513–528.
• K.-W. Chen, C.-L. Chung, and M. Eie, “Sum formulas of multiple zeta values with arguments multiples of a common positive integer”, J. Number Theory 177 (2017), 479–496.
• M. Eie, Topics in number theory, Monographs in Number Theory 2, World Scientific, 2009.
• M. Eie, The theory of multiple zeta values with applications in combinatorics, Monographs in Number Theory 7, World Scientific, 2013.
• M. E. Hoffman, “Multiple harmonic series”, Pacific J. Math. 152:2 (1992), 275–290.
• M. E. Hoffman, “The algebra of multiple harmonic series”, J. Algebra 194:2 (1997), 477–495.
• M. E. Hoffman, “On multiple zeta values of even arguments”, Int. J. Number Theory 13:3 (2017), 705–716.
• Y. Ohno and N. Wakabayashi, “Cyclic sum of multiple zeta values”, Acta Arith. 123:3 (2006), 289–295.
• D. Zagier, “Evaluation of the multiple zeta values $\zeta(2,\ldots,2,3,2,\ldots,2)$”, Ann. of Math. $(2)$ 175:2 (2012), 977–1000.