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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 \(L(s, \chi)\)
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