Stieltjes constants

The Stieltjes constants are a sequence of real numbers that are determined by the limit ${\displaystyle \gamma _{n}}$

${\displaystyle \gamma _{n}:=\lim _{N\to \infty }\left(\sum _{k=1}^{N}{\frac {\log ^{n}k}{k}}-{\frac {\log ^{n+1}N}{n+1}}\right),\quad n=0,1,2,\dotsc }$

are defined, where is Euler's constant . It is believed that they are irrational. Proof of this has not yet been found. Because of their definition, they are sometimes referred to as generalized Euler's constants . They appear in the Laurent expansion of the Riemann zeta function ${\displaystyle \gamma _{0}}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma _{n}}$

${\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}\gamma _{n}}{n!}}(s-1)^{n}}$

and when evaluating certain specific integrals on:

${\displaystyle \int \limits _{0}^{\infty }{\frac {\log ^{2}x}{e^{x}+1}}\,\mathrm {d} x=(\log 2)\,{\big (}{\frac {1}{3}}\log ^{2}2+\zeta (2)-\gamma ^{2}-2\gamma _{1}{\big )}=1{,}121192486\dots }$

They are closely related to the numbers

${\displaystyle \tau _{n}:=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {\log ^{n}k}{k}},\quad n=0,1,2,\dotsc }$

together. These can be calculated numerically well using an acceleration of convergence (continued averaging). The recursion applies

${\displaystyle \tau _{0}=\log 2}$

${\displaystyle \tau _{n}={\frac {\log ^{n+1}2}{n+1}}-\sum _{k=0}^{n-1}{\binom {n}{k}}\log ^{n-k}2\cdot \gamma _{k},\qquad n=1,2,\dotsc }$

and the explicit representation with the help of Bernoulli's numbers :

${\displaystyle \gamma _{n}=-{\frac {1}{n+1}}\sum _{k=0}^{n+1}{\binom {n+1}{k}}B_{n+1-k}\log ^{n-k}2\cdot \tau _{k},\quad n=0,1,2,\dotsc }$

From the recursion we get for the identity , i. H. for Euler's constant the alternating series ${\displaystyle n=1}$ ${\displaystyle \tau _{1}={\tfrac {1}{2}}\log ^{2}2-\gamma \log 2}$

${\displaystyle \gamma ={\frac {1}{2}}\log 2+{\frac {1}{\log 2}}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\log k}{k}}={\frac {1}{2}}\log 2+\sum _{k=2}^{\infty }(-1)^{k}{\frac {\log _{2}k}{k}},}$

which is very similar to Vacca's range .

The result shows an oscillating behavior with the "frequency" falling asymptotically slowly towards 0. It is known that ${\displaystyle \gamma _{n}}$

${\displaystyle \limsup _{n\to \infty }{\frac {\ln |\gamma _{n}|}{n}}=\ln \ln n}$

applies.

Numerical values

n	Decimal expansion of γ n	OEIS
0	0.577215664901532860606512090082 ...	A001620
1	−0.0728158454836767248605863758749 ...	A082633
2	−0.00969036319287231848453038603521 ...	A086279
3	0.00205383442030334586616004654275 ...	A086280
4th	0.00232537006546730005746817017752 ...	A086281
5	0.000793323817301062701753334877444 ...	A086282
6th	−0.000238769345430199609872421841908 ...	A183141
7th	−0.000527289567057751046074097505478 ...	A183167
8th	−0.000352123353803039509602052165001 ...	A183206
9	−0.000034394774418088048177914623798 ...	A184853
10	0.000205332814909064794683722289237 ...	A184854

generalization

The following is important for the Hurwitz zeta function :

${\displaystyle \gamma _{n}(a):=\lim _{N\to \infty }\left(\sum _{k=1}^{N}{\frac {\log ^{n}(k+a)}{k+a}}-{\frac {\log ^{n+1}(N+a)}{n+1}}\right),\quad n=0,1,2,\dotsc }$

literature

• Rick Kreminski: Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants. In: Mathematics of Computation. V. 72, no. 243, 2003, pp. 1379-1397.
• Charles Knessl, Mark W, Coffey: An effective asymptotic formula for the Stieltjes Constants. In: Mathematics of Computation. V. 80, no. 273, 2010, pp. 379-386.

Web links

• Eric W. Weisstein : Stieltjes Constants . In: MathWorld (English).
• Values ​​of the Stieltjes constants from 0 to 78, each with 256 decimal places