An efficient algorithm for the Hurwitz zeta and related functions (Q1005995)

The author proves propositions which yield efficient algorithms for computing the Riemann and Hurwitz zeta functions in \({\mathbb C}\) and extend an algorithm of \textit{P. Borwein} [Can. Math. Soc. Conf. Proc. 27, 29--34 (2000; Zbl 0984.11067)]. His sharpest proposition is the following Theorem. Let \[ \varepsilon_n(s)={1\over {L_n(-1)(1-2^{1-s}})}{1\over {\Gamma (s)}}\int_0^1 {L_n(x)\over {x+1}}(-\ln x)^{s-1} \,dx \] with the Laguerre polynomial \[ L_n(x)=\sum_{j=0}^n {n \choose n-j}{{(-x)^j}\over{ j!}} \] and \[ c_j=(-1)^j\Big (\sum_{k=0}^j {n \choose n-k}{1\over {k!}} -L_n(-1)\Big ). \] Then there is an effectively computable constant \(C>0\) such that for \(\sigma=\text{Re}(s) > 0\) \[ \zeta(s)=-{{1\over {L_n(-1)(1-2^{1-s})}}\sum_{j=0}^{n-1}{{c_j}\over{(j+1)^s}}} + \varepsilon_n(s), \leqno (\#) \] \[ |\varepsilon_n(s)|\leq Cn^{1/4}e^{-2\sqrt n}{(1+|t|/\sigma)e^{\pi |t|/2}\over {|1-2^{1-s}|}} \] with \(t=\text{Im}(s)\), as \(n\to \infty\). A generalization of the identity (\#) is given for \(\zeta(s,a)\) with \(\text{Re}(s)>1\), \(\text{Re}(a)>0\), for \(\zeta(s,{{a+1}\over {2}}) - \zeta(s,{{a}\over {2}})\) with \(\text{Re}(s)>0, \text{Re}(a)>0\) and for the Lerch zeta-function \(\Phi(z,s,a)\) with \(\text{Re}(a)>0\) for \(\text{Re}(s)>0\) when \(|z|\leq 1\) but \(z\neq 1\) or for \(\text{Re}(s)>1\) when \(|z|=1\). Let \(a\in {\mathbb C}\smallsetminus ({\mathbb Z}\smallsetminus \mathbb N)\). For \(s\in {\mathbb C}\) when \(|z|<1\) or for \(\text{Re}(s)>1\) when \(|z|=1\) \[ \Phi(z,s,a)=\sum_{n=0}^{\infty} {{z^n}\over {(n+a)^s}}. \] Brief numerical examples are presented.