Minakshisundaram–Pleijel zeta function

Remarks

By the Minakshisundaram–Pleijel expansion of the heat trace and the formula $\Gamma(s) Z(s) = \int_0^\infty t^{s-1} \big( \sum_{j \ge 0} e^{-t\lambda_j} - 1 \big) \, dt$, the function $Z(s)$ has a meromorphic continuation to $\C$. Its poles are simple and lie among the points $s = n/2 - k$, $k = 0, 1, 2, \ldots$, other than $0, -1, -2, \ldots$; in particular $Z$ is holomorphic at $s = 0$. The residue at $s = n/2$ is $\operatorname{vol}(M) / \big( (4\pi)^{n/2} \Gamma(n/2) \big)$, which is equivalent to Weyl's law.

Example: for $M = \R / 2\pi\Z$ the nonzero eigenvalues are $k^2$, $k \ge 1$, each with multiplicity $2$. So $Z(s) = 2 \sum_{k \ge 1} k^{-2s} = 2 \zeta(2s)$, with $\zeta$ the Riemann zeta function. It has a single simple pole at $s = 1/2$ with residue $1$, and $Z(0) = 2\zeta(0) = -1$.

Non-example: the sum $\sum_{j \ge 0} \lambda_j^{-s}$ that includes $\lambda_0 = 0$ is not defined, since $0^{-s}$ has no meaning; this is why the zero eigenvalue is left out.