Stieltjes constants

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In mathematics, the Stieltjes constants are the numbers γk 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 \rightarrow \infty} {\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}

(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!}{2\pi} \int_0^{2\pi} e^{-nix} \zeta\left(e^{ix}+1\right) dx.

The zero'th constant γ0 = γ = 0.577... is known as the Euler-Mascheroni constant.

The first few values are:

n approximate value of γn
0 0.5772156649015328606065120900824024310421
1 -0.072815845483676724860586
2 -0.0096903631928723184845303
3 0.002053834420303345866160
4 0.0023253700654673000574
5 0.0007933238173010627017
6 -0.00023876934543019960986
7 -0.0005272895670577510
8 -0.00035212335380
9 -0.0000343947744
10 0.000205332814909

More generally, one can define Stieltjes constants γk(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\;

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