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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

\( \gamma_n = \lim_{m \rightarrow \infty} {\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}. \)

(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 \( \gamma_0 = \gamma = 0.577\dots \) is known as the Euler–Mascheroni constant.

The first few values are:

n approximate value of γn OEIS
0 0.5772156649015328606065120900824024310421 A001620
1 -0.072815845483676724860586 A082633
2 -0.0096903631928723184845303 A086279
3 0.002053834420303345866160 A086280
4 0.0023253700654673000574 A086281
5 0.0007933238173010627017 A086282
6 -0.00023876934543019960986 A183141
7 -0.0005272895670577510 A183167
8 -0.00035212335380 A183206
9 -0.0000343947744 A184853
10 0.000205332814909 A184854

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

\( \gamma_n(1)=\gamma_n.\; \)

References

Weisstein, Eric W., "Stieltjes Constants" from MathWorld.
Plouffe, Simon. "Stieltjes Constants, from 0 to 78, 256 digits each".
Kreminski, Rick (2003). "Newton-Cotes integration for approximating Stieltjes generalized Euler constants". Mathematics of Computation 72 (243): 1379–1397. doi:10.1090/S0025-5718-02-01483-7. MR 1972742.
Coffey, Mark W. (2009). "Series representations for the Stieltjes constants". arXiv:0905.1111.
Coffey, Mark W. (2010). "Addison-type series representation for the Stieltjes constants". J. Number Theory 130: 2049–2064. doi:10.1016/j.jnt.2010.01.003. MR 2653214.
Knessl, Charles; Coffey, Mark W. (2011). "An effective asymptotic formula for the Stieltjes constants". Math. Comp. 80 (273): 379–386. doi:10.1090/S0025-5718-2010-02390-7. MR 2728984.

When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

Mathematics Encyclopedia

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