## Rapidly convergent series representations for $$\zeta(2n+1)$$ and their $$\chi$$-analogue.(English)Zbl 0933.11042

The aim of this paper is to bring forth two rapidly convergent series representations for $$\zeta(2n+1)$$ and an analogue for the Dirichlet series $$L(s,\chi)$$. In Theorem 1 the author obtains the identity \begin{aligned} \zeta(2n+1) - n&\sum_{l=1}^\infty{\cos(2\pi lx)\over{\l}^{2n+1}} - \pi x\sum_{l=1}^\infty{\sin(2\pi lx)\over{\l}^{2n}}\\ =(-1)^n(2\pi x)^{2n}&\left\{\sum_{k=1}^{n-1}(-1)^{k-1}{k\zeta(2k+1) \over(2n-2k)!(2\pi x)^{2k}} + \sum_{k=0}^\infty {(2k)!\zeta(2k) \over(2n+2k)!}x^{2k}\right\},\end{aligned}\tag{1} where $$n$$ is a positive integer and $$x$$ a real variable satisfying $$|x|\leq 1$$. In the case when $$x = {1\over 2}$$ one obtains the recent identity of Cvijović-Klinowski [Proc. Am. Math. Soc. 125, 1263-1271 (1997; Zbl 0863.11055)], which generalizes a classical identity of Euler, reproved recently by J. A. Ewell [see, e.g., Rocky Mt. J. Math. 25, 1003-1012 (1995; Zbl 0851.11049)]. Besides (1), the author proves another identity for $$\zeta(2n+1)$$, plus the analogue for $$L(2n+1,\chi)$$, where $$\chi$$ is a primitive character for a modulus $$q$$. The proof of (1) and the other results is by clever use of a contour integral, the functional equation for $$\zeta(s)$$, and evaluation of certain integrals.
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M35 Hurwitz and Lerch zeta functions