Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta function (Q1998904)

This interesting paper constructs sequences of complex numbers that rapidly converge to the Hurwitz zeta function \[ \zeta(s,a)=\sum_{k=0}^{\infty}\,\frac{1}{(k+a)^s},\ \Re(a)>0\text{ and }\Re(s)>1. \] The method uses diagonal Padé approximants to the remainder series \[\sum_{k=n}^{\infty}\,\frac{1}{(k+a)^s}.\] An important tool is is the series \[ \Phi_s(z)=\sum_{n=0}^{\infty}\,\frac{(s)_{2n+1}}{(2n+2)!}B_{2n+2}(-z)^n\, z\not=0\text{ and }s>0, \] with \((B_{2n+2})_{n\geq 0}\) the Bernoulli numbers, which is the asymptotic expansion of \[ \hat{\Phi}_s(z)=\int_0^{\infty}\,\frac{\mu_s(x)}{1-zx}\,dx, \] with \(\mu_s(x)=\omega_s(\sqrt{x})/2\sqrt{x}\in L^1(\mathbb{R}^{+}\) for \(\Re(s)>0\) and \[ \omega_s(x)=\frac{2(-1)^mx^s}{\Gamma(s)\Gamma(m+1-s)}\,\int_x^{\infty}\,(t-x)^{m-s}\frac{d^m}{dt^m}\left(\frac{1}{e^{2\pi t}-1}\right)\,dt. \] The main results are: \textbf{Theorem 1.} (\S1) Let \(s>0,\,s\not= 1\) and \(a\in\mathbb{C}\) such that \(\Re(a)>0\). Set \(a_n=n+a\). Then, for every large enough integer \(n\) and any integer \(k\geq 1\), we have \[ \zeta(s,a)=\sum_{j=0}^{n-1}\frac{1}{(j+a)^s}+\frac{1}{(s-1)a_n^{s-1}}+\frac{1}{2a_n^s}+\frac{1}{a_n^{s+1}}[k/k]_{\Phi_s}\left(-\frac{1}{a_n^2}\right)+ \varepsilon_{k,s} \left(\frac{1}{a_n^2}\right), \] where \[ | \varepsilon_{k,s} (1/a_n^2)|\leq D_s\frac{(2k+2\rho)\Gamma(2k+\rho+1)^2}{|a_n|^{4k+2}(4k+2\rho+1)(2k+1)\left(\begin{matrix}4k+2\rho\\ 2k+1\end{matrix}\right)^2}, \] where \(\rho=(m+7)/2\) and \(D_s=(2\pi)^sm!/\gamma(s)\) and \(m=[s]\). \textbf{Corollary 1.} (\S1) Let \(r\in\mathbb{Q}\) such that \(0<r<2e\). Let \(s>0,s\not= 1\). Then, for every integer \(n\geq 1\) such that \(rn\) is an integer, we have \[ \zeta(s)=\sum_{k=1}^n\frac{1}{k^s}+\frac{1}{(s-1)n^{s-1}}-\frac{1}{2n^s}+\frac{1}{n^{s+1}}[rn/rn]-{\Phi_s}\left(-\frac{1}{n^2}\right)+\delta_{r,s,n}, \] where \[ \limsup_{n\rightarrow\infty}\,|\delta_{r,s,n}|^{1/n}\leq\left(\frac{r}{2e}\right)^{4r}. \] \textbf{Proposition 1.} (\S3) For any \(s>0\) and any \(x\geq 0\), we have \[ 0<\Gamma(s)x\omega_s(x)\leq 2(2\pi)^{s-1}m!G\left(\frac{m+5}{2},1,x\right), \] where \(m=[s]\) and \[ G(\alpha,\beta,x)=|\Gamma(\alpha+ix)\Gamma(\beta+ix)|^2. \] The layout of the paper is as follows: \S1. Introduction (\(\frac{1}{2}\) pages) \S2. Consequences of an integral representation of \(\zeta(s,a)\) (\(1\frac{1}{2}\) pages) \S3. Bounds for the weight \(\omega_s(x)\) (\(3\) pages) \S4. Wilson's polynomials (\(1\frac{1}{2}\) pages) \S5. A bound for the Padé approximant of \(\Phi_s(z)\) (\(2\) pages) \S6. Proofs of Theorem 1 and Corollary 1 (\(\frac{1}{2}\) page) \S7. The case \(s\) real (negative) In this section the main results given above are generalized to the case \(s<0\) \S8. The case \(a=1\) and \(s\in\mathbb{N}\) (\(1\) page) References (\(10\) items)