## 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)

### MSC:

 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.)
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### References:

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