Author:
CELIK SERMIN CAM, ,GORAL HAYDAR,
Abstract
In this note, we study the evaluations of Euler sums via trigonometric series. It is a commonly believed conjecture that for an even weight greater than seven, Euler sums cannot be evaluated in terms of the special values of the Riemann zeta function. For an even weight, we reduce the evaluations of Euler sums into the evaluations of double series and integrals of products of Clausen functions. We also re-evaluate Euler sums of odd weight using a new method based on trigonometric series.
Subject
Applied Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis
Reference11 articles.
1. [1] E. Alkan and H. G¨oral, Trigonometric series and special values of L-functions. J. Number Theory 178 (2017), 94-117.
2. [2] D. Borwein, J.M. Borwein, and R. Girgensohn, Explicit evaluation of Euler sums. Proc. Edinburgh Math. Soc. (2), 38 (1995), 277-294.
3. [3] K. Boyadzhiev, Evaluation of Euler-Zagier sums. Int. J. Math. Math. Sci. 27 (2001), 7, 407-412.
4. [4] D.M. Bradley and X. Zhou, On Mordell-Tornheim sums and multiple zeta values. Ann. Sci. Math. Qu'ebec 34 (2010), 1, 15-23.
5. [5] R. Harada, On Euler's formulae for double zeta values. Kyushu J. Math. 72 (2018), 15-24.