| 释义 |
Stieltjes ConstantsN.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Expanding the Riemann Zeta Function about  gives
 | (1) |
 | (2) |
 | (3) |
 gives the Euler-Mascheroni Constant  . The first few numerical values are givenin the following table. |  | | 0 | 0.5772156649 | | 1 |  | | 2 |  | | 3 | 0.002053834420 | | 4 | 0.002325370065 | | 5 | 0.0007933238173 |
Briggs (1955-1956) proved that there infinitely many of each Sign. Berndt (1972) gave upper bounds of
 | (4) |
 | (5) |
 is the Floor Function. Hardy (1912) gave the Formula
 | (6) |
 .
A set of constants related to is
 | (7) |
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. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/stltjs/stltjs.html Hardy, G. H. ``Note on Dr. Vacca's Series for .'' Quart. J. Pure Appl. Math. 43, 215-216, 1912. 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. 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. Sitaramachandrarao, R. ``Maclaurin Coefficients of the Riemann Zeta Function.'' Abstracts Amer. Math. Soc. 7, 280, 1986. Vacca, G. ``A New Series for the Eulerian Constant.'' Quart. J. Pure Appl. Math. 41, 363-368, 1910.
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