Formulas for higher derivatives of the Riemann zeta function
HTML articles powered by AMS MathViewer
 by Tom M. Apostol PDF
 Math. Comp. 44 (1985), 223232 Request permission
Abstract:
The functional equation for $\zeta (s)$ is used to obtain formulas for all derivatives ${\zeta ^{(k)}}(s)$. A closed form evaluation of ${\zeta ^{(k)}}(0)$ is given, and numerical values are computed to 15D for $k = 0(1)18$.References

Milton Abramowitz & Irene A. Stegun, Handbook of Mathematical Functions, Nat. Bur. Standards, Appl. Math. Series No. 55, Washington, D. C., 1964.
Tom M. Apostol, Calculus, Vol. II, 2nd ed., Wiley, New York, 1969.
 Tom M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, SpringerVerlag, New YorkHeidelberg, 1976. MR 0434929 B. Baillaud & H. Bourget, Correspondence d’Hermite et de Stieltjes, Tome I, GauthierVillars, Paris, 1905.
 Bruce C. Berndt, The number of zeros for $\zeta ^{(k)}\,(s)$, J. London Math. Soc. (2) 2 (1970), 577–580. MR 266874, DOI 10.1112/jlms/2.Part_{4}.577
 Bruce C. Berndt, Chapter 8 of Ramanujan’s second notebook, J. Reine Angew. Math. 338 (1983), 1–55. MR 684013, DOI 10.1515/crll.1983.338.1
 L. Bourguet, Sur les intégrales eulériennes et quelques autres fonctions uniformes, Acta Math. 2 (1883), no. 1, 261–295 (French). MR 1554599, DOI 10.1007/BF02415217 Henry M. Jeffery, "On the derivatives of the gammafunction," Quart. J. Math., v. 6, 1864, pp. 82108.
 J. J. Y. Liang and John Todd, The Stieltjes constants, J. Res. Nat. Bur. Standards Sect. B 76B (1972), 161–178. MR 326974 Niels Nielsen, Handbuch der Theorie der Gammafunktion, Teubner, Leipzig, 1906.
 Robert Spira, Zerofree regions of $\zeta ^{(k)}(s)$, J. London Math. Soc. 40 (1965), 677–682. MR 181621, DOI 10.1112/jlms/s140.1.677
 Robert Spira, Another zerofree region for $\zeta ^{(k)}\,(s)$, Proc. Amer. Math. Soc. 26 (1970), 246–247. MR 263754, DOI 10.1090/S00029939197002637544
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, Oxford, at the Clarendon Press, 1951. MR 0046485
Additional Information
 © Copyright 1985 American Mathematical Society
 Journal: Math. Comp. 44 (1985), 223232
 MSC: Primary 11M06
 DOI: https://doi.org/10.1090/S00255718198507710445
 MathSciNet review: 771044