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 $z=1$ gives

$$\zeta(z)={1\over z-1}+\sum_{n=0}^\infty {(-1)^n\over n!}\gamma_n(z-1)^n,$$ (1)

where

$$\gamma_n\equiv\lim_{m\to\infty}\left[{\,\sum_{k=1}^m {(\ln k)^n\over k}-{(\ln m)^{n+1}\over n+1}}\right].$$ (2)

An alternative definition is given by

$$\gamma_n'\equiv {(-1)^n\over n!}\gamma_n.$$ (3)

The case $n=0$ gives the Euler-Mascheroni Constant $\gamma$. The first few numerical values are given in the following table.

$n$	$\gamma_n$
0	0.5772156649
1	$-0.07281584548$
2	$-0.009690363192$
3	0.002053834420
4	0.002325370065
5	0.0007933238173

Briggs (1955-1956) proved that there infinitely many $\gamma_n$ of each Sign. Berndt (1972) gave upper bounds of

$$\vert\gamma_n\vert<\cases{ {4(n-1)!\over \pi^n} & for $n$\ even\cr {2(n-1)!\over \pi^n} & for $n$\ odd.\cr}$$ (4)

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

$$\gamma=\sum_{k=1}^\infty {(-1)^k\over k}\left\lfloor{\lg k}\right\rfloor ,$$ (5)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function. Hardy (1912) gave the Formula

$${2\gamma_1\over\ln 2}=\sum_{k=1}^\infty {(-1)^k\over k}[2\lg... ...oor{\lg (2k)}\right\rfloor ]\left\lfloor{\lg k}\right\rfloor .$$ (6)

Kluyver (1927) gave similar series for $\gamma_n$ with $n>1$.

A set of constants related to $\gamma_n$ is

$$\delta_n\equiv\lim_{m\to\infty}\left[{\sum_{k=1}^m (\ln k)^n-\int_1^m (\ln x)^n\,dx-{\textstyle{1\over 2}}(\ln m)^n}\right]$$ (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 $\gamma$.'' 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.