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Laurent series coefficients of the logarithmic derivative of the Riemann zeta function at 1
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Numbers
$n$
$n$th coefficient
INPUT{numbers.yaml} (not shown in preview)
Definition
This list contains the coefficients $\eta_n$ of the Laurent series expansion $\zeta'(s)/\zeta(s) = - \sum_{n=-1}^\infty \eta_n (s-1)^n$. We additionally list $\eta_n/n$, the coefficients in (3).
Parameters
$n$
—   integer ($n \geq -1$)
Formulas
(1)
$\eta_0 = -\gamma$, where $\gamma=\gamma_0$ is the Euler-Mascheroni constant.
(2)
$\eta_n = -(-1)^n \frac{n+1}{n!}\gamma_n + \sum_{k=0}^{n-1} \frac{(-1)^{n-k}}{(n-k-1)!}\eta_k \gamma_{n-k-1}$ for $n\geq 0$, where $\gamma_n$ are the Stieltjes constants. [1]
(3)
$\log \zeta(s) = -\log(s-1) - \sum_{n=1}^\infty \frac{\eta_n}{n} (s-1)^n$.
The radius of convergence of the expansion of $\zeta'(s)/\zeta(s)$ at $s=1$ is $3$, due to the trivial zero of $\zeta(s)$ at $s=-2$.
J. B. Keiper, "Power series expansions of Riemann's $\xi$ function", Math. Comp. 58 (1992), 765-773. (doi)