Stieltjes constants

The Stieltjes constants $\scriptstyle \gamma _{n}\,$ for $\scriptstyle n\,\in \,\mathbb {N} _{0}\,$ arise from the Laurent expansion of $\scriptstyle \zeta (z)\,$ about $\scriptstyle z\,=\,1\,$, where $\scriptstyle \zeta (z)\,$ is the Riemann zeta function. There are infinitely many positive as well as negative Stieltjes constants. These constants are named after Thomas Jan Stieltjes. The Stieltjes constants are sometimes referred to as generalized Euler constants.

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

where $\scriptstyle \gamma \,$ is the Euler-Mascheroni constant.

$\gamma _{n}=\lim _{m\to \infty }{\Bigg [}\sum _{k=1}^{m}{\frac {(\log k)^{n}}{k}}-\int _{1}^{m}{\frac {(\log x)^{n}}{x}}dx{\Bigg ]}=\lim _{m\to \infty }{\Bigg [}\sum _{k=1}^{m}{\frac {(\log k)^{n}}{k}}-{\frac {(\log m)^{n+1}}{n+1}}{\Bigg ]}\,$

Euler-Mascheroni constant

Since

$\lim _{s\to 1}{\bigg [}\zeta (s)-{\frac {1}{s-1}}{\bigg ]}=\gamma \,$

it implies that $\scriptstyle \gamma _{0}\,=\,\gamma \,$.

$\scriptstyle \gamma _{0}\,$ is the familiar Euler-Mascheroni constant usually notated just $\scriptstyle \gamma \,$.

$\gamma =\gamma _{0}=\lim _{m\to \infty }{\Bigg [}\sum _{k=1}^{m}{\frac {1}{k}}-\int _{1}^{m}{\frac {1}{x}}dx{\Bigg ]}=\lim _{m\to \infty }{\Bigg [}\sum _{k=1}^{m}{\frac {1}{k}}-\log m{\Bigg ]}\,$

Table of Stieltjes constants

Stieltjes constants

$n\,$	$\gamma _{n}\,$ (8 places)	A-number
0	+ 0.57721566	A001620
1	− 0.07281584	A082633 × (−1)
2	− 0.00969036	A086279 × (−1)
3	+ 0.00205383	A086280
4	+ 0.00232537	A086281
5	+ 0.00079332	A086282
6	− 0.00023876	A183141 × (−1)
7	− 0.00052728	A183167 × (−1)
8	− 0.00035212	A183206 × (−1)
9	− 0.00003439	A184853 × (−1)
10	+ 0.00020533	A184854

Inequality

$|\gamma _{n}|<{\frac {2(n-1)!}{\pi ^{n}}},\,n\geq 1.\,$

External links

• Weisstein, Eric W., Stieltjes Constants, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/StieltjesConstants.html]
• Mark W. Coffey, Series of zeta values, the Stieltjes constants, and a sum S_\gamma(n), 2009.
• Simon Plouffe, Stieltjes Constants from 0 to 78, to 256 digits each. Copyright: Simon Plouffe/Plouffe's Inverter (c) 1986.