Stieltjes constants

In mathematics, the Stieltjes constants are the numbers $gamma\_k$ that occur in the Laurent series expansion of the Riemann zeta function:

:$zeta(s)=frac\{1\}\{s-1\}+sum\_\{n=0\}^infty\; frac\{(-1)^n\}\{n!\}\; gamma\_n\; ;\; (s-1)^n.$

The Stieltjes constants are given by the limit

: $gamma\_n\; =\; lim\_\{m\; ightarrow\; infty\}\{left(left(sum\_\{k\; =\; 1\}^m\; frac\{(ln\; k)^n\}\{k\}\; ight)\; -\; frac\{(ln\; m)^\{n+1\{n+1\}\; ight)\}.$

(In the case "n" = 0, the first summand requires evaluation of 0⁰, which is taken to be 1.)

Cauchy's differentiation formula leads the integral representation

:$gamma\_n\; =\; frac\{(-1)^n\; n!\}\{2pi\}\; int\_0^\{2pi\}\; e^\{-nix\}\; zetaleft(e^\{ix\}+1\; ight)\; dx.$

The zero'th constant $gamma\_0\; =\; gamma\; =\; 0.577...$ is known as the Euler-Mascheroni constant.

The first few values are:

:::

More generally, one can define Stieltjes constants $gamma\_k(q)$ that occur in the Laurent series expansion of the Hurwitz zeta function:

:$zeta(s,q)=frac\{1\}\{s-1\}+sum\_\{n=0\}^infty\; frac\{(-1)^n\}\{n!\}\; gamma\_n(q)\; ;\; (s-1)^n.$

Here "q" is a complex number with Re("q")>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have:$gamma\_n(1)=gamma\_n.;$

ee also

* Euler-Mascheroni constant

References

*

* Plouffe's inverter. [http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt Stieltjes Constants, from 0 to 78, 256 digits each]