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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

$|\gamma_n| < \frac{2 (n-1)!}{\pi^n},\, n \ge 1. \,$