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Stieltjes Constants

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Expanding the Riemann zeta function about z=1 gives

zeta(z)=1/(z-1)+sum_(n=0)^infty((-1)^n)/(n!)gamma_n(z-1)^n
(1)

(Havil 2003, p. 118), where the constants

gamma_n=lim_(m->infty)[sum_(k=1)^m((lnk)^n)/k-((lnm)^(n+1))/(n+1)]
(2)

are known as Stieltjes constants.

Another sum that can be used to define the constants is

zeta(z+1)-1/z=sum_(k=0)^infty((-1)^kgamma_kz^k)/(k!).
(3)

These constants are returned by the Wolfram Language function StieltjesGamma[n].

A generalization gamma_n(a) takes gamma_n(a)/n! as the coefficient of (1-s)^n is the Laurent series of the Hurwitz zeta function zeta(s,a) about s=1. These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].

The case n=0 gives the usual Euler-Mascheroni constant

gamma_0=gamma.
(4)

A limit formula for gamma_1 is given by

gamma_1=-lim_(y->infty)y{y+I[zeta(1+i/y)]},
(5)

where I[z] is the imaginary part and zeta(z) is the Riemann zeta function.

An alternative definition is given by absorbing the coefficient of gamma_n into the constant,

gamma_n^'=((-1)^n)/(n!)gamma_n
(6)

(e.g., Hardy 1912, Kluyver 1927).

The Stieltjes constants are also given by

gamma_n=lim_(z->1)[(-1)^nzeta^((n))(z)-(n!)/((z-1)^(n+1))].
(7)
StieltjesGamma
StieltjesGammaLog

Plots of the values of the Stieltjes constants as a function of n are illustrated above (Kreminski). The first few numerical values are given in the following table.

nOEISgamma_n
0A0016200.5772156649
1A082633-0.07281584548
2A086279-0.009690363192
3A0862800.002053834420
4A0862810.002325370065
5A0862820.0007933238173
StieltjesGammaSignRuns

Briggs (1955-1956) proved that there infinitely many gamma_n of each sign. The signs of gamma_n for n=0, 1, ... are 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, ... (OEIS A114523), and the run lengths of consecutive signs are 1, 2, 3, 4, 3, 4, 5, 4, 5, 5, 5, ... (OEIS A114524). A plot of run lengths is shown above.

Berndt (1972) gave upper bounds of

|gamma_n|<{(4(n-1)!)/(pi^n) for n even; (2(n-1)!)/(pi^n) for n odd.
(8)

However, these bounds are extremely weak. A stronger bound is given by

|gamma_n|<10^(-4)e^(nlnlnn)
(9)

for n>4 (Matsuoka 1985).

Vacca (1910) proved that the Euler-Mascheroni constant may be expressed as

gamma=sum_(k=1)^infty((-1)^k)/k|_lgk_|,
(10)

where |_x_| is the floor function and the lg function lgx=log_2x is the logarithm to base 2. Hardy (1912) derived the formula

gamma_1=1/6(ln2)^2-1/2gammaln2+1/(2ln2)sum_(k=1)^infty((-1)^k(lnk)^2)/k
(11)

from Vacca's expression.

Kluyver (1927) gave similar series for gamma_n valid for all n>1,

gamma_n=n!(ln2)^nsum_(m=1)^(n+1)((-1)^(m-1))/(m!)sum_(k=1)^infty((-1)^k|_lgk_|^mB_(1+n-m)(lgk))/k,
(12)

where B_n(x) is a Bernoulli polynomial. However, this series converges extremely slowly, requiring more than 10^4 terms to get two digits of gamma_1 and many more for higher order gamma_n.

gamma_n can also be expressed as a single sum using

gamma_n=((ln2)^n)/(n+1)sum_(k=1)^infty((-1)^k)/kB_(n+1)(lgk).
(13)

gamma_1 also appears in the asymptotic expansion of the sum

sum_(n=1)^x1/nln(x/n)=1/2(lnx)^2+gammalnx-gamma_1+O(x^(-1)),
(14)

where gamma_1 was called -D and given incorrectly by Ellision and Mendès-France (1975) (and the error was subsequently reproduced by Le Lionnais 1983, p. 47). The exact form of (14) is given by

sum_(n=1)^x1/nln(x/n)=H_xlnx+gamma_1(x+1)-gamma_1,
(15)

where H_x is a harmonic number and gamma_n(a) is a generalized Stieltjes constant.

A set of constants related to gamma_n is

delta_n=lim_(m->infty)[sum_(k=1)^m(lnk)^n-int_1^m(lnx)^ndx-1/2(lnm)^n]
(16)

(Sitaramachandrarao 1986, Lehmer 1988).

The Stieltjes constants also satisfy the beautiful sum

sum_(k=0)^infty(gamma_(k+n))/(k!)=(-1)^n[n!+zeta^((n))(0)]
(17)

(O. Marichev, pers. comm., 2008).

See also

Bernoulli Polynomial, Euler-Mascheroni Constant, Euler Product, Riemann Zeta Function

Related Wolfram sites

http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/StieltjesGamma/

Explore with Wolfram|Alpha

References

Berndt, B. C. "On the Hurwitz Zeta-Function." Rocky Mountain J. Math. 2, 151-157, 1972.Bohman, J. and Fröberg, C.-E. "The Stieltjes Function--Definitions and Properties." Math. Comput. 51, 281-289, 1988.Briggs, W. E. "Some Constants Associated with the Riemann Zeta-Function." Mich. Math. J. 3, 117-121, 1955-1956.Coffey, M. W. "New Results on the Stieltjes Constants: Asymptotic and Exact Evaluation." J. Math. Anal. Appl. 317, 603-612, 2006.Coffey, M. W. "New Summation Relations for the Stieltjes Constants." Proc. Roy. Soc. A 462, 2563-2573, 2006.Ellison, W. J. and Mendès-France, M. Les nombres premiers. Paris: Hermann, 1975.Finch, S. R. "Stieltjes Constants." §2.21 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 166-171, 2003.Hardy, G. H. "Note on Dr. Vacca's Series for gamma." Quart. J. Pure Appl. Math. 43, 215-216, 1912.Hardy, G. H. and Wright, E. M. "The Behavior of zeta(s) when s->1." §17.3 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 246-247, 1979.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Kluyver, J. C. "On Certain Series of Mr. Hardy." Quart. J. Pure Appl. Math. 50, 185-192, 1927.Knopfmacher, J. "Generalised Euler Constants." Proc. Edinburgh Math. Soc. 21, 25-32, 1978.Kreminski, R. "Newton-Cotes Integration for Approximating Stieltjes (Generalized Euler) Constants." Math. Comput. 72, 1379-1397, 2003.Kreminski, R. "This Site Will Archive Some Stieltjes-Related Computational Work..." http://www.tamu-commerce.edu/math/FACULTY/KREMIN/stieltjesrelated/.Kreminski, R. "This Page Displays Work in Progress by Rick Kreminski." http://www.tamu-commerce.edu/math/FACULTY/KREMIN/stieltjes/stieltjestestpage.html.Kreminski, R. "Gammas 1 to 12 to 6900 Digits." http://www.tamu-commerce.edu/math/FACULTY/KREMIN/stieltjesrelated/gammas1to12/.Lammel, E. "Ein Beweis dass die Riemannsche Zetafunktion zeta(s) is |s-1|<=1 keine Nullstelle besitzt." Univ. Nac. Tucmán Rev. Ser. A 16, 209-217, 1966.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 47, 1983.Lehmer, D. H. "The Sum of Like Powers of the Zeros of the Riemann Zeta Function." Math. Comput. 50, 265-273, 1988.Liang, J. J. Y. and Todd, J. "The Stieltjes Constants." J. Res. Nat. Bur. Standards--Math. Sci. 76B, 161-178, 1972.Matsuoka, Y. "Generalized Euler Constants Associated with the Riemann Zeta Function." In Number Theory and Combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984). Singapore: World Scientific, pp. 279-295, 1985.Plouffe, S. "Stieltjes Constants from 0 to 78, to 256 Digits Each." http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt.Sitaramachandrarao, R. "Maclaurin Coefficients of the Riemann Zeta Function." Abstracts Amer. Math. Soc. 7, 280, 1986.Sloane, N. J. A. Sequences A001620/M3755, A082633, A086279, A086280, A086281, A086282, A114523, and A114524 in "The On-Line Encyclopedia of Integer Sequences."Vacca, G. "A New Series for the Eulerian Constant." Quart. J. Pure Appl. Math. 41, 363-368, 1910.

Referenced on Wolfram|Alpha

Stieltjes Constants

Cite this as:

Weisstein, Eric W. "Stieltjes Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StieltjesConstants.html

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