Stieltjes constants

In mathematics, the Stieltjes constants are the numbers γ_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 γ_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". http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt. 
• 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. MR1972742. 
• 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. MR2653214. 
• 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. MR2728984.