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Grenz-Formel

An equation derived by Kronecker:

sum^'_(x,y,z=-infty)^infty(x^2+y^2+dz^2)^(-s)=4zeta(s)eta(s)+(2pi)/(s-1)(zeta(2s-2))/(d^(s-1)) +(2pi^s)/(Gamma(s))d^((1-s)/2)sum_(n=1)^inftyn^((s-1)/2)sum_(u^2|n)(r(n/(u^2)))/(u^(2s-2))int_0^inftye^(pisqrt(nd)(y+y^(-1)))y^(s-2)dy,

where r(n) is the sum of squares function, 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 terms with zero denominator (Selberg and Chowla 1967).

See also

Dirichlet Eta Function, Epstein Zeta Function, Lattice Sum, Sum of Squares Function

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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.

Referenced on Wolfram|Alpha

Grenz-Formel

Cite this as:

Weisstein, Eric W. "Grenz-Formel." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Grenz-Formel.html

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