Rocky Mountain Journal of Mathematics

Series representations for the Stieltjes constants

Mark W. Coffey

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The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for $\gamma_k=\gamma_k(1)$. Some extensions are briefly described, as well as the relevance to expansions of Dirichlet $L$ functions.

Article information

Rocky Mountain J. Math., Volume 44, Number 2 (2014), 443-477.

First available in Project Euclid: 4 August 2014

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Zentralblatt MATH identifier

Primary: 11M35: Hurwitz and Lerch zeta functions 11M06: $\zeta (s)$ and $L(s, \chi)$ 11Y60: Evaluation of constants
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]

Stieltjes constants Riemann zeta function Hurwitz zeta function Laurent expansion Stirling numbers of the first kind Dirichlet L functions Lerch zeta function


Coffey, Mark W. Series representations for the Stieltjes constants. Rocky Mountain J. Math. 44 (2014), no. 2, 443--477. doi:10.1216/RMJ-2014-44-2-443.

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  • M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, National Bureau of Standards, Washington, 1964.
  • T.M. Apostol, Introduction to analytic number theory, Springer Verlag, New York, 1976; corrected fourth printing, 1995.
  • P. Appell, Sur la nature arithmétique de la constante d'Euler, Comp. Rend. Acad. Sci. 182 (1926), 897–899, 949.
  • B.C. Berndt, On the Hurwitz zeta function, Rocky Mountain J. Math. 2 (1972), 151–157.
  • W.E. Briggs, Some constants associated with the Riemann zeta-function, Michigan Math. J. 3 (1955), 117–121.
  • M.W. Coffey, Functional equations for the Stieltjes constants, arXiv:1402.3746, 2014.
  • ––––, Certain logarithmic integrals, including solution of monthly problem $11629$, zeta values, and expressions for the Stieltjes constants, arXiv:1201.3393, 2012.
  • ––––, The Stieltjes constants, their relation to the $\eta_j$ coefficients, and representation of the Hurwitz zeta function, Analysis 30 (2010), 383–409.
  • ––––, Series of zeta values, the Stieltjes constants, and a sum $S_\gamma(n)$, arXiv/math-ph/:0706.0345v2, 2009.
  • ––––, On some series representations of the Hurwitz zeta function, J. Comp. Appl. Math. 216 (2008), 297–305.
  • ––––, On representations and differences of Stieltjes coefficients, and other relations, Rocky Mountain J. Math. 41 (2011), 1815–1846, arXiv:0809.3277.
  • ––––, Theta and Riemann xi function representations from harmonic oscillator eigenfunctions, Phys. Lett. 362 (2007), 352–356.
  • ––––, New results on the Stieltjes constants: Asymptotic and exact evaluation, J. Math. Anal. Appl. 317 (2006), 603–612..
  • ––––, Relations and positivity results for derivatives of the Riemann $\xi$ function, J. Comp. Appl. Math. 166 (2004), 525–534.
  • L. Comtet, Advanced combinatorics, D. Reidel, Dordrecht, 1974.
  • H.M. Edwards, Riemann's zeta function, Academic Press, New York, 1974.
  • N.J. Fine, Note on the Hurwitz zeta-function, Proc. Amer. Math. Soc. 2 (1951), 361–364.
  • I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Academic Press, New York, 1980.
  • R.L. Graham, D.E. Knuth and O. Patashnik, Concrete mathematics, 2nd ed., Addison Wesley, New York, 1994.
  • G.H. Hardy, Note on Dr. Vacca's series for $\gamma$, Quart. J. Pure Appl. Math. 43 (1912), 215–216.
  • A. Ivić, The Riemann zeta-function, Wiley, New York, 1985.
  • A.A. Karatsuba and S.M. Voronin, The Riemann zeta-function, Walter de Gruyter, New York, 1992.
  • J. Katriel, Stirling number identities: Interconsistency of $q$-analogues, J. Phys. 31 (1998), 3559–3572.
  • D. Klusch, On the Taylor expansion of the Lerch zeta-function, J. Math. Anal. Appl. 170 (1992), 513–523.
  • J.C. Kluyver, On certain series of Mr. Hardy, Quart. J. Pure Appl. Math. 50 (1927), 185–192.
  • C. Knessl, private communication, 2009.
  • C. Knessl and M.W. Coffey, An asymptotic form for the Stieltjes constants $\gamma_k(a)$ and for a sum $S_\gamma(n)$ appearing under the Li criterion, Math. Comp. 80 (2011), 2197–2217.
  • R. Kreminski, Newton-Cotes integration for approximating Stieltjes $($generalized Euler$)$ constants, Math. Comp. 72 (2003), 1379–1397.
  • K. Maślanka, A hypergeometric-like representation of zeta-function of Riemann, arXiv/math-ph/:0105007v1, 2001.
  • D. Mitrović, The signs of some constants associated with the Riemann zeta function, Michigan Math. J. 9 (1962), 395–397.
  • B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. Preuss. Akad. Wiss. 671 (1859–1860).
  • J. Ser, Inter. Math. (1925), 126–127.
  • R. Smith, private communication, 2008.
  • T.J. Stieltjes, Correspondance d'Hermite et de Stieltjes, Volumes 1 and 2, Gauthier-Villars, Paris, 1905.
  • E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., Oxford University Press, Oxford, 1986.
  • E.W. Weisstein, Stieltjes constants, from MathWorld–A Wolfram Web Resource,
  • H.S. Wilf, The asymptotic behavior of the Stirling numbers of the first kind, J. Comb. Theor. 64 (1993), 344–349.
  • J.R. Wilton, A note on the coefficients in the expansion of $\zeta(s,x)$ in powers of $s-1$, Quart. J. Pure Appl. Math. 50 (1927), 329–332.
  • N.-Y. Zhang and K.S. Williams, Some results on the generalized Stieltjes constants, Analysis 14 (1994), 147–162.