nLab
Epstein zeta function

Definition

Let τ=τ 1+iτ 2\tau = \tau_1 + i\tau_2\in\mathbb{C} with τ 2>0\tau_2\gt 0. Then, the corresponding Epstein Zeta Function is a meromorphic function which appears by analytic continuation to the complex plane of the function ζ τ:sζ τ(s)\zeta_\tau:s\to\zeta_\tau(s) which is for Re(s)>0Re(s)\gt 0 defined by the formula

ζ τ(s)= (n,m)(0,0)n,m(τ 2|n+mτ|) s \zeta_\tau(s) = \sum_{\overset{n,m\in\mathbb{Z}}{(n,m)\neq(0,0)}}\left(\frac{\tau_2}{|n+m\tau|}\right)^s

Bibliography

  • for related notions see zeta function and the links therein
  • L. Kronecker, Zur Theorie der Elliptischen Functionen I, IV (1883) in Leopold Kronecker’s Werke IV
  • C. L. Siegel, Lectures on advanced analytic number theory, Tata Institute (1961)

Created on January 6, 2018 at 07:43:36. See the history of this page for a list of all contributions to it.