# Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x$ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far]

For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as $$f(x) = \zeta(1-x) + 1/x \qquad x \in \mathbb N$$ and then the scaled derivatives $$g(x,d) = x^d \cdot {f^{(d)}(x)\over d!}$$.

In Pari/GP there is a very nice procedure for the computation of $\zeta$ implemented, but I don't see anything for the derivatives except to implement numerical differentiation along the line ${f(x+\epsilon/2) - f(x-\epsilon/2)\over \epsilon}$ and so on, but which becomes inaccurate for higher derivatives.
For small x I can use the representation of the function $f(x)$ as a power series involving the Stieltjes-constants themselves, but a suitable computation is then possible only for a small range of x-values if I use only, say 100 or 200 terms of the power series.

Q: What would be a more efficient representation of $f(x)$ and/or $g(x)$ for the actual computation, where -say- x and d both can go up to 100 or 200 ? Note, that for high c (the index for the partial sums, see below) there is the significant effect , that either (x is small and d is high) or converse, such that d+x in the terms of one partial sum is constant.

More context: what I'm finally after is to be able to explore the partial sums and its single terms in
$$S \underset{\mathfrak E}{=} \lim_{c\to \infty} S(c) = \sum_{k=0}^{c} (-1)^k g( c+1-k ,k)$$ which should be a representation for the (divergent) sum of the Stieltjes constants, taken by some "Eulerian" summation procedure $\mathfrak E$ (see an earlier post here on MO).

I have implemented simultaneous computation of $(\zeta(s,a), \zeta'(s,a) \ldots, \zeta^{(n)}(s,a))$ for arbitrary $s, a \in \mathbb{C}$, with the option to subtract $1/(s-1)$ to remove the singularity at $s = 1$, in my library for ball arithmetic Arb. All the derivatives are computed with rigorous error bounds (as currently described on this page).
As a benchmark, computing the first 1000 Stieltjes constants to 1000 accurate digits (using 1500 digits of precision to give $\gamma_0$ accurate to 1500 digits and $\gamma_{1000}$ accurate to 1000 digits) takes 14 seconds, and computing the first 5000 Stieltjes constants to 5000 digits takes 40 minutes.
• Yes (if I understand your question directly), the Euler-Maclaurin formula works directly for zeta at any integer, and gives you the Stieltjes constants at $s = 1$ after just removing the singular $1/(s-1)$ term. – Fredrik Johansson Mar 8 '13 at 20:59