epstein zeta function

Input interpretation

Epstein zeta function

Definition

The Epstein zeta function for a n×n matrix S of a positive definite real quadratic form and ρ a complex variable with ℜ[ρ]>n/2 (where ℜ[z] denotes the real part) is defined by Z_n(S, ρ) = 1/2 sum'_(a element Z^n) (a^T S a)^(-ρ), where the sum is over all column vectors with integer coordinates and the prime means the summation excludes the origin. Epstein derived the analytic continuation, functional equation, and so-called Kronecker limit formula for this function.
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Related terms

Grenz-Formel | Riemann zeta function | zeta function

Subject classifications

MathWorld

Riemann zeta function | point lattices

MSC 2010

05C35 | 11Mxx
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