Grenz-Formel

An equation derived by Kronecker:

$\setbox0=\hbox{$\scriptstyle{}$}\setbox2=\hbox{$\displaystyle{\sum}$}\setbox4=\... ... (x^2+y^2+dz^2)^{-s}=4\zeta(s)\eta(s)+{2\pi\over s-1}{\zeta(2s-2)\over d^{s-1}}$
$+{2\pi^s\over\Gamma(s)} d^{(1-s)/2}\sum_{n=1}^\infty n^{(s-1)/2} \sum_{u^2\ver... ...2}\right)\over u^{2s-2}} \int_0^\infty e^{\pi\sqrt{nd}\,(y+y^{-1})}y^{s-2}\,dy,$

where

$$r(n)=4\sum_{d\vert n} \sin({\textstyle{1\over 2}}\pi d),$$

$\zeta(z)$, is the Riemann Zeta Function, $\eta(z)$ is the Dirichlet Eta Function, $\Gamma(z)$ is the Gamma Function, and the primed sum omits infinite terms (Selberg and Chowla 1967).

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 296-297, 1987.

Selberg, A. and Chowla, S. ``On Epstein's Zeta-Function.'' J. Reine. Angew. Math. 227, 86-110, 1967.