Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta function. (English) Zbl 1475.11152

This interesting paper constructs sequences of complex numbers that rapidly converge to the Hurwitz zeta function \[ \zeta(s,a)=\sum_{k=0}^{\infty}\,\frac{1}{(k+a)^s},\ \operatorname{Re}(a)>0\text{ and }\operatorname{Re}(s)>1. \] 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 \(\operatorname{Re}(s)>0\) 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:
Theorem 1. (§1) Let \(s>0,\,s\not= 1\) and \(a\in\mathbb{C}\) such that \(\operatorname{Re}(a)>0\). 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 \(m=[s]\).
Corollary 1. (§1) 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 \[ \limsup_{n\rightarrow\infty}\,|\delta_{r,s,n}|^{1/n}\leq\left(\frac{r}{2e}\right)^{4r}. \]
Proposition 1. (§3) 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:
§1. Introduction (\(\frac{1}{2}\) pages)
§2. Consequences of an integral representation of \(\zeta(s,a)\) (\(1\frac{1}{2}\) pages)
§3. Bounds for the weight \(\omega_s(x)\) (\(3\) pages)
§4. Wilson’s polynomials (\(1\frac{1}{2}\) pages)
§5. A bound for the Padé approximant of \(\Phi_s(z)\) (\(2\) pages)
§6. Proofs of Theorem 1 and Corollary 1 (\(\frac{1}{2}\) page)
§7. The case \(s\) real (negative)
In this section the main results given above are generalized to the case \(s<0\)
§8. The case \(a=1\) and \(s\in\mathbb{N}\) (\(1\) page)
References (\(10\) items)


11M35 Hurwitz and Lerch zeta functions
11Y60 Evaluation of number-theoretic constants
41A21 Padé approximation
11J72 Irrationality; linear independence over a field
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI HAL


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