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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

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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)

$\log \zeta(s) = -\log(s-1) - \sum_{n=1}^\infty \frac{\eta_n}{n} (s-1)^n$.

Comments

(4)

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 $s=-2$.

References

[1]

M.W. Coffey, "Relations and positivity results for the derivatives of the Riemann ξ function", J. Comput. Appl. Math. 166 (2004), 525-534.

[2]

Donal F. Connon, "A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants", (2009), 50 pages. (arXiv)

[3]

J. B. Keiper, "Power series expansions of Riemann's $\xi$ function", Math. Comp. 58 (1992), 765-773. (doi)

Links

Data properties

Entries are of type: real number

Table is complete: no