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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 \(L(s, \chi)\)

Citations:

Zbl 0147.305
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Digital Library of Mathematical Functions:

(25.11.19) ‣ s -Derivatives ‣ §25.11(vi) Derivatives ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.20) ‣ s -Derivatives ‣ §25.11(vi) Derivatives ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.6) ‣ §25.11(iii) Representations by the Euler–Maclaurin Formula ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11(iii) Representations by the Euler–Maclaurin Formula ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11(vi) Derivatives ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.18(i) Function Values and Derivatives ‣ §25.18 Methods of Computation ‣ Computation ‣ Chapter 25 Zeta and Related Functions
(25.4.5) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.4.6) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.11) ‣ §25.6(ii) Derivative Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.12) ‣ §25.6(ii) Derivative Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.13) ‣ §25.6(ii) Derivative Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.14) ‣ §25.6(ii) Derivative Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.6(ii) Derivative Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions