## Stieltjes Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Expanding the Riemann Zeta Function about gives

 (1)

where
 (2)

An alternative definition is given by
 (3)

The case gives the Euler-Mascheroni Constant . The first few numerical values are given in 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)

Vacca (1910) proves that the Euler-Mascheroni Constant may be expressed as
 (5)

where is the Floor Function. Hardy (1912) gave the Formula
 (6)

Kluyver (1927) gave similar series for with .

A set of constants related to is

 (7)

(Sitaramachandrarao 1986, Lehmer 1988).

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.