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Stieltjes Constants
المؤلف:
Berndt, B. C.
المصدر:
"On the Hurwitz Zeta-Function." Rocky Mountain J. Math. 2
الجزء والصفحة:
...
4-2-2020
2252
Stieltjes Constants
Expanding the Riemann zeta function about gives
![]() |
(1) |
(Havil 2003, p. 118), where the constants
![]() |
(2) |
are known as Stieltjes constants.
Another sum that can be used to define the constants is
![]() |
(3) |
These constants are returned by the Wolfram Language function StieltjesGamma[n].
A generalization takes
as the coefficient of
is the Laurent series of the Hurwitz zeta function
about
. These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].
The case gives the usual Euler-Mascheroni constant
![]() |
(4) |
A limit formula for is given by
(5) |
where is the imaginary part and
is the Riemann zeta function.
An alternative definition is given by absorbing the coefficient of into the constant,
![]() |
(6) |
(e.g., Hardy 1912, Kluyver 1927).
The Stieltjes constants are also given by
![]() |
(7) |
Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the following table.
![]() |
OEIS | ![]() |
0 | A001620 | 0.5772156649 |
1 | A082633 | ![]() |
2 | A086279 | ![]() |
3 | A086280 | 0.002053834420 |
4 | A086281 | 0.002325370065 |
5 | A086282 | 0.0007933238173 |
Briggs (1955-1956) proved that there infinitely many of each sign. The signs of
for
, 1, ... are 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
(8) |
However, these bounds are extremely weak. A stronger bound is given by
![]() |
(9) |
for (Matsuoka 1985).
Vacca (1910) proved that the Euler-Mascheroni constant may be expressed as
![]() |
(10) |
where is the floor function and the lg function
is the logarithm to base 2. Hardy (1912) derived the formula
![]() |
(11) |
from Vacca's expression.
Kluyver (1927) gave similar series for valid for all
,
![]() |
(12) |
where is a Bernoulli polynomial. However, this series converges extremely slowly, requiring more than
terms to get two digits of
and many more for higher order
.
can also be expressed as a single sum using
![]() |
(13) |
also appears in the asymptotic expansion of the sum
![]() |
(14) |
where was called
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
![]() |
(15) |
where is a harmonic number and
is a generalized Stieltjes constant.
A set of constants related to is
![]() |
(16) |
(Sitaramachandrarao 1986, Lehmer 1988).
The Stieltjes constants also satisfy the beautiful sum
![]() |
(17) |
(O. Marichev, pers. comm., 2008).
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 ." Quart. J. Pure Appl. Math. 43, 215-216, 1912.
Hardy, G. H. and Wright, E. M. "The Behavior of when
." §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 is
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.
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