Stieltjes constants

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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 zero'th constant $\gamma _{0}=\gamma =0.577\dots$ is known as the Euler–Mascheroni constant.

Representations

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

The first few values are:

n	approximate value of γ_n	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 .

Asymptotic growth

The Stieltjes constants satisfy the bound

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

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

$|\gamma _{n}|<0.0001e^{{n\log \log n}}$

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

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

Weisstein, Eric W., "Stieltjes Constants", 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.
Johansson, Fredrik (2013). "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives". arXiv:1309.2877.

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