ON THE EPSTEIN ZETA FUNCTION

Abstract

The Epstein zeta function $Z(s)$ is defined for Re$s > 1$ by

\[Z(s)=\sum_{m,n=-\infty, (m, n)\neq (0, 0)}^\infty \frac{1}{(am^2−bnm+cn^2)^s}\]

where $a$, $b$, $c$ are real numbers with $a>0$ and $b^2 - 4ac<0$. $Z(s)$ can be continued analytically to the whole complex plane except for a simple pole at $s =1$. Simple proofs of the functional equation and of the Kronecker "Grenz-formel" for $Z (s)$ are given. The value of $Z(k)$($k =2, 3, \cdots$) is determined in terms of infinite series of the form

\[\sum_{n=1}^\infty\frac{\cot^r n\pi\tau}{n^{2k-1}} (r=1, 2, \cdots, k)\]

where $\tau=(b+\sqrt{b^2-4ac})/2a$.