---
tag: T1039Q9FY0
kind: concept
name: Minakshisundaram–Pleijel zeta function
msc: [58J50, 11M41, 58J52]
sources:
  - url: https://en.wikipedia.org/wiki/Minakshisundaram–Pleijel_zeta_function
same_as:
  wikidata: Q11342341
  wikipedia: Minakshisundaram–Pleijel zeta function
---

## Summary

The Minakshisundaram–Pleijel zeta function of a closed Riemannian manifold is the Dirichlet series $\sum \lambda_j^{-s}$ over the nonzero eigenvalues of its Laplacian.
It has a pointwise version, with each term weighted by eigenfunctions.
It converges for $\operatorname{Re} s > n/2$ and continues to a meromorphic function on $\C$.
Its poles and special values carry geometric information, and its derivative at $0$ defines the [[zeta-regularized determinant|YT3ZQ2F7FG]] of the Laplacian.
It generalizes the [[Riemann zeta function|1XWBY4N0VT]], which arises from the circle.

## Explanations

### L1 human

The eigenvalues of the Laplacian on a closed manifold are the squared frequencies of its vibrations.
Summing their negative powers gives a function of $s$, built like the Riemann zeta function from the positive integers.
For the circle of length $2\pi$ the eigenvalues are $1, 1, 4, 4, 9, 9, \ldots$, and the zeta function is $2\zeta(2s)$.
In general one cannot write it down, but its analytic properties follow from the heat kernel.