Fine Art


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

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


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

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World