Stieltjes constants

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The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.

In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:

The constant is known as the Euler–Mascheroni constant.


The Stieltjes constants are given by the limit

(In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.)

Cauchy's differentiation formula leads to the integral representation

Various representations in terms of integrals and infinite series are given in works of Jensen, Franel, Hermite, Hardy, Ramanujan, Ainsworth, Howell, Coppo, Connon, Coffey, Choi, Blagouchine and some other authors.[1][2][3][4][5][6] In particular, Jensen-Franel's integral formula, often erroneously attributed to Ainsworth and Howell, states that

where δn,k is the Kronecker symbol (Kronecker delta).[5][6] Among other formulae, we find


As concerns series representations, a famous series implying an integer part of a logarithm was given by Hardy in 1912[8]

Israilov[9] gave semi-convergent series in terms of Bernoulli numbers

Connon,[10] Blagouchine[6][11] and Coppo[1] gave several series with the binomial coefficients

where Gn are Gregory's coefficients, also known as reciprocal logarithmic numbers (G1=+1/2, G2=−1/12, G3=+1/24, G4=−19/720,... ). More general series of the same nature include these examples[11]



where ψn(a) are the Bernoulli polynomials of the second kind and Nn,r(a) are the polynomials given by the generating equation

respectively (note that Nn,1(a) = ψn(a)).[12] Oloa and Tauraso[13] showed that series with harmonic numbers may lead to Stieltjes constants

Blagouchine[6] obtained slowly-convergent series involving unsigned Stirling numbers of the first kind

as well as semi-convergent series with rational terms only

where m=0,1,2,... In particular, series for the first Stieltjes constant has a surprisingly simple form

where Hn is the nth harmonic number.[6] More complicated series for Stieltjes constants are given in works of Lehmer, Liang, Todd, Lavrik, Israilov, Stankus, Keiper, Nan-You, Williams, Coffey.[2][3][6]

Bounds and asymptotic growth[edit]

The Stieltjes constants satisfy the bound

given by Berndt in 1972.[14] Better bounds in terms of elementary functions were obtained by Lavrik[15]

by Israilov[9]

with k=1,2,... and C(1)=1/2, C(2)=7/12,... , by Nan-You and Williams[16]

by Blagouchine[6]

where Bn are Bernoulli numbers, and by Matsuoka[17][18]

As concerns estimations resorting to non-elementary functions and solutions, Knessl, Coffey[19] and Fekih-Ahmed[20] obtained quite accurate results. For example, Knessl and Coffey give the following formula that approximates the Stieltjes constants relatively well for large n.[19] If v is the unique solution of

with , and if , then


Up to n = 100000, the Knessl-Coffey approximation correctly predicts the sign of γn with the single exception of n = 137.[19]

Numerical values[edit]

The first few values are:

n approximate value of γn OEIS
0 +0.5772156649015328606065120900824024310421593359 A001620
1 −0.0728158454836767248605863758749013191377363383 A082633
2 −0.0096903631928723184845303860352125293590658061 A086279
3 +0.0020538344203033458661600465427533842857158044 A086280
4 +0.0023253700654673000574681701775260680009044694 A086281
5 +0.0007933238173010627017533348774444448307315394 A086282
6 −0.0002387693454301996098724218419080042777837151 A183141
7 −0.0005272895670577510460740975054788582819962534 A183167
8 −0.0003521233538030395096020521650012087417291805 A183206
9 −0.0000343947744180880481779146237982273906207895 A184853
10 +0.0002053328149090647946837222892370653029598537 A184854
100 −4.2534015717080269623144385197278358247028931053 × 1017
1000 −1.5709538442047449345494023425120825242380299554 × 10486
10000 −2.2104970567221060862971082857536501900234397174 × 106883
100000 +1.9919273063125410956582272431568589205211659777 × 1083432

For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.

Further information related to the numerical evaluation of Stieltjes constants may be found in works of Keiper,[21] Kreminski,[22] Plouffe,[23] Johansson[24][25] and Blagouchine.[25] First, Johansson provided values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each (the numerical values can be retrieved from the LMFDB [1]. Later, Johansson and Blagouchine devised a particularly efficient algorithm for computing generalized Stieltjes constants (see below) for large n and complex a, which can be also used for ordinary Stieltjes constants.[25] In particular, it allows one to compute γn to 1000 digits in a minute for any n up to n=10100.

Generalized Stieltjes constants[edit]

General information[edit]

More generally, one can define Stieltjes constants γn(a) that occur in the Laurent series expansion of the Hurwitz zeta function:

Here a is a complex number with Re(a)>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γn(1)=γn The zero'th constant is simply the digamma-function γ0(a)=-Ψ(a),[26] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them. For example, there exists the following asymptotic representation

due to Berndt and Wilton. The analog of Jensen-Franel's formula for the generalized Stieltjes constant is the Hermite formula[5]

Similar representations are given by the following formulas:[25]


Generalized Stieltjes constants satisfy the following recurrence relation

as well as the multiplication theorem

where denotes the binomial coefficient (see[27] and,[28] pp. 101–102).

First generalized Stieltjes constant[edit]

The first generalized Stieltjes constant has a number of remarkable properties.

  • Malmsten's identity (reflection formula for the first generalized Stieltjes constants): the reflection formula for the first generalized Stieltjes constant has the following form

where m and n are positive integers such that m<n. This formula has been long-time attributed to Almkvist and Meurman who derived it in 1990s.[29] However, it was recently reported that this identity, albeit in a slightly different form, was first obtained by Carl Malmsten in 1846.[5][30]

  • Rational arguments theorem: the first generalized Stieltjes constant at rational argument may be evaluated in a quasi-closed form via the following formula

see Blagouchine.[5][26] An alternative proof was later proposed by Coffey[31] and several other authors.

  • Finite summations: there are numerous summation formulae for the first generalized Stieltjes constants. For example,

For more details and further summation formulae, see.[5][28]

  • Some particular values: some particular values of the first generalized Stieltjes constant at rational arguments may be reduced to the gamma-function, the first Stieltjes constant and elementary functions. For instance,

At points 1/4, 3/4 and 1/3, values of first generalized Stieltjes constants were independently obtained by Connon[32] and Blagouchine[28]

At points 2/3, 1/6 and 5/6

These values were calculated by Blagouchine.[28] To the same author are also due

Second generalized Stieltjes constant[edit]

The second generalized Stieltjes constant is much less studied than the first constant. Similarly to the first generalized Stieltjes constant, the second generalized Stieltjes constant at rational argument may be evaluated via the following formula

see Blagouchine.[5] An equivalent result was later obtained by Coffey by another method.[31]


  1. ^ a b c Marc-Antoine Coppo. Nouvelles expressions des constantes de Stieltjes. Expositiones Mathematicae, vol. 17, pp. 349-358, 1999.
  2. ^ a b Mark W. Coffey. Series representations for the Stieltjes constants, arXiv:0905.1111
  3. ^ a b Mark W. Coffey. Addison-type series representation for the Stieltjes constants. J. Number Theory, vol. 130, pp. 2049-2064, 2010.
  4. ^ Junesang Choi. Certain integral representations of Stieltjes constants, Journal of Inequalities and Applications, 2013:532, pp. 1-10
  5. ^ a b c d e f g h Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009. And vol. 151, pp. 276-277, 2015. arXiv:1401.3724
  6. ^ a b c d e f g Iaroslav V. Blagouchine. Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. Corrigendum: vol. 173, pp. 631-632, 2017. arXiv:1501.00740
  7. ^ "A couple of definite integrals related to Stieltjes constants". Stack Exchange.
  8. ^ G. H. Hardy. Note on Dr. Vacca's series for γ, Q. J. Pure Appl. Math. 43, pp. 215–216, 2012.
  9. ^ a b M. I. Israilov. On the Laurent decomposition of Riemann's zeta function [in Russian]. Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 158, pp. 98-103, 1981.
  10. ^ Donal F. Connon Some applications of the Stieltjes constants, arXiv:0901.2083
  11. ^ a b Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 18A (#A3): 1–45
  12. ^ Actually Blagouchine gives more general formulas, which are valid for the generalized Stieltjes constants as well.
  13. ^ "A closed form for the series ..." Stack Exchange.
  14. ^ Bruce C. Berndt. On the Hurwitz Zeta-function. Rocky Mountain Journal of Mathematics, vol. 2, no. 1, pp. 151-157, 1972.
  15. ^ A. F. Lavrik. On the main term of the divisor's problem and the power series of the Riemann's zeta function in a neighbourhood of its pole (in Russian). Trudy Mat. Inst. Akad. Nauk. SSSR, vol. 142, pp. 165-173, 1976.
  16. ^ Z. Nan-You and K. S. Williams. Some results on the generalized Stieltjes constants. Analysis, vol. 14, pp. 147-162, 1994.
  17. ^ Y. Matsuoka. Generalized Euler constants associated with the Riemann zeta function. Number Theory and Combinatorics: Japan 1984, World Scientific, Singapore, pp. 279-295, 1985
  18. ^ Y. Matsuoka. On the power series coefficients of the Riemann zeta function. Tokyo Journal of Mathematics, vol. 12, no. 1, pp. 49-58, 1989.
  19. ^ a b c Charles Knessl and Mark W. Coffey. An effective asymptotic formula for the Stieltjes constants. Math. Comp., vol. 80, no. 273, pp. 379-386, 2011.
  20. ^ Lazhar Fekih-Ahmed. A New Effective Asymptotic Formula for the Stieltjes Constants, arXiv:1407.5567
  21. ^ J.B. Keiper. Power series expansions of Riemann ζ-function. Math. Comp., vol. 58, no. 198, pp. 765-773, 1992.
  22. ^ Rick Kreminski. Newton-Cotes integration for approximating Stieltjes generalized Euler constants. Math. Comp., vol. 72, no. 243, pp. 1379-1397, 2003.
  23. ^ Simon Plouffe. Stieltjes Constants, from 0 to 78, 256 digits each
  24. ^ Fredrik Johansson. Rigorous high-precision computation of the Hurwitz zeta function and its derivatives, arXiv:1309.2877
  25. ^ a b c d Johansson, Fredrik; Blagouchine, Iaroslav (2019), "Computing Stieltjes constants using complex integration", Mathematics of Computation, 88 (318): 1829–1850, arXiv:1804.01679, doi:10.1090/mcom/3401
  26. ^ a b "Definite integral". Stack Exchange.
  27. ^ Donal F. Connon New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions, arXiv:0903.4539
  28. ^ a b c d Iaroslav V. Blagouchine Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. The Ramanujan Journal, vol. 35, no. 1, pp. 21-110, 2014. Erratum-Addendum: vol. 42, pp. 777-781, 2017. PDF
  29. ^ V. Adamchik. A class of logarithmic integrals. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.
  30. ^ "Evaluation of a particular integral". Stack Exchange.
  31. ^ a b Mark W. Coffey Functional equations for the Stieltjes constants, arXiv:1402.3746
  32. ^ Donal F. Connon The difference between two Stieltjes constants, arXiv:0906.0277