## Formulas for higher derivatives of the Riemann zeta function.(English)Zbl 0557.10029

The paper contains a new formula for $$(-1)^ k \zeta^{(k)}(1-s)$$, which enables the author to determine explicitly the coefficients $$a_{jkm}$$ and $$b_{jkm}$$ of a previous formula of a similar kind due to R. Spira [J. Lond. Math. Soc. 40, 677-682 (1965; Zbl 0147.305)]. As a consequence, a closed form evaluation of $$\zeta^{(k)}(0)$$ is obtained. The results about $$\zeta^{(k)}(0)$$ contain the well known formulae when $$k=1,2$$, and appear to be new if $$k\geq 3$$. Numerical values are given for $$k=1,...,18$$.
Reviewer: A.Perelli

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$

Zbl 0147.305
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