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Fast approximations of certain number-theoretic constants. (English. Russian original) Zbl 1361.11084
Dokl. Math. 91, No. 3, 283-286 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 462, No. 2, 137-140 (2015).
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 \[ P_m=\prod_{q=1}^m \sum_{_{\substack{ j_q=j_{q+1}+1 \\ j_{m+1}=0 }}}^{n-q} \frac 1{(n-j_q)^2};\quad m=1, 2,\ldots, k-1;\quad P_0=1, \]
\[ D_m=d_m=(-1)^m; \quad m=1, 2,\ldots, k-2; \]
\[ D_{k-1}=(-1)^{k-1}\left(1+\frac 14\cdot \frac{2n+1}{2n-1}\right),\quad d_{k-1}=(-1)^{k-1}\cdot \frac 54, \]
\[ Z_k=1+\frac 1{2^{2k}}+\frac 1{3^{2k}}+\ldots +\frac 1{(k-1)^{2k}},\quad k\geq 2, \quad Z_1=0, \]
\[ \tilde Z_k=1+\frac 1{2^{2k+1}}+\frac 1{3^{2k+1}}+\ldots +\frac 1{(k-1)^{2k+1}},\quad k\geq 2, \quad \tilde Z_1=0. \]
MSC:
11Y60 Evaluation of number-theoretic constants
11Y35 Analytic computations
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
65D15 Algorithms for approximation of functions
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References:
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