Stieltjes constants

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The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant.

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 zero'th constant \gamma_0 = \gamma = 0.577\dots is known as the Euler–Mascheroni constant.

Representations[edit]

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 to the integral representation

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

Several representations in terms of integrals and infinite series are given in the papers of Coffey.

Numerical values[edit]

The first few values are:

n approximate value of γn OEIS
0 +0.5772156649015328606065120900824024310421593359 A001620
1 −0.0728158454836767248605863758749013191377363383 A082633
2 −0.0096903631928723184845303860352125293590658061 A086279
3 +0.0020538344203033458661600465427533842857158044 A086280
4 +0.0023253700654673000574681701775260680009044694 A086281
5 +0.0007933238173010627017533348774444448307315394 A086282
6 −0.0002387693454301996098724218419080042777837151 A183141
7 −0.0005272895670577510460740975054788582819962534 A183167
8 −0.0003521233538030395096020521650012087417291805 A183206
9 −0.0000343947744180880481779146237982273906207895 A184853
10 +0.0002053328149090647946837222892370653029598537 A184854
100 −4.2534015717080269623144385197278358247028931053 × 1017
1000 −1.5709538442047449345494023425120825242380299554 × 10486
10000 −2.2104970567221060862971082857536501900234397174 × 106883
100000 +1.9919273063125410956582272431568589205211659777 × 1083432

For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern.

Numerical values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each, have been computed by Johansson. The numerical values can be retrieved from the LMFDB [1].

Asymptotic growth[edit]

The Stieltjes constants satisfy the bound

|\gamma_n| < \frac{4 (n - 1)!}{{\pi}^n,}

as proved by Berndt. A much tighter bound, valid for n \ge 10, is given by Matsuoka:

|\gamma_n| < 0.0001 e^{n \log \log n}

Knessl and Coffey give a formula that approximates the Stieltjes constants accurately for large n. If v is the unique solution of

2 \pi \exp(v \tan v) = n \frac{\cos(v)}{v}

with 0 < v < \pi/2, and if u = v \tan v, then

\gamma_n \sim \frac{B}{\sqrt{n}} e^{nA} \cos(an+b)

where

A = \frac{1}{2} \log(u^2+v^2) - \frac{u}{u^2+v^2}
B = \frac{2 \sqrt{2\pi} \sqrt{u^2+v^2}}{[(u+1)^2+v^2]^{1/4}}
a = \tan^{-1}\left(\frac{v}{u}\right) + \frac{v}{u^2+v^2}
b = \tan^{-1}\left(\frac{v}{u}\right) - \frac{1}{2} \left(\frac{v}{u+1}\right).

Up to n = 10^5, the Knessl-Coffey approximation correctly predicts the sign of \gamma_n with the single exception of n = 137.

Generalized Stieltjes constants[edit]

More generally, one can define Stieltjes constants \gamma_k(a) that occur in the Laurent series expansion of the Hurwitz zeta function:

\zeta(s,a)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(a) \; (s-1)^n.

Here a is a complex number with Re(a)>0. Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have

\gamma_n(1)=\gamma_n.\;

References[edit]