nLab Epstein zeta function

Definition

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

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

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)