Fast approximations of certain number-theoretic constants
From the text: In the present paper the problem of construction of fast approximations of zeta constants by rational fractions is solved. The author proves: Theorem 1. For any integer \(k\), \(k\geq 1\), the following relations for the values of the Riemann zeta-function of even and odd argument are valid: \[ \zeta(2k)=Z_k+2\sum_{n=k}^\infty \frac{(-1)^{n-1}}{\binom{2n}{n}n^2} \sum_{m=0}^{k-1}\frac{D_mP_m}{n^{2k-2m-2}}, \] \[ \zeta(2k+1)=\tilde Z_k+2\sum_{n=k}^\infty \frac{(-1)^{n-1}} {\binom{2n}{n} n^3} \sum_{m=0}^{k-1}\frac{d_mP_m}{n^{2k-2m-2}}. \] Here \[ P_m=\prod_{q=1}^m \sum_{_{\substack{ j_q=j_{q+1}+1 \\ j_{m+1}=0 }}}^{n-q} \frac 1{(n-j_q)^2};\quad m=1, 2,\ldots, k-1;\quad P_0=1, \] \[ D_m=d_m=(-1)^m; \quad m=1, 2,\ldots, k-2; \] \[ D_{k-1}=(-1)^{k-1}\left(1+\frac 14\cdot \frac{2n+1}{2n-1}\right),\quad d_{k-1}=(-1)^{k-1}\cdot \frac 54, \] \[ Z_k=1+\frac 1{2^{2k}}+\frac 1{3^{2k}}+\ldots +\frac 1{(k-1)^{2k}},\quad k\geq 2, \quad Z_1=0, \] \[ \tilde Z_k=1+\frac 1{2^{2k+1}}+\frac 1{3^{2k+1}}+\ldots +\frac 1{(k-1)^{2k+1}},\quad k\geq 2, \quad \tilde Z_1=0. \]
fast approximations of zeta constants by rational fractions
Riemann zeta-function of even and odd argument
scientific article; zbMATH DE number 6485663