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.

Grenz-Formel | Riemann zeta function | zeta function