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Introduction

$\displaystyle S^1=\mathbb{R}/2\pi \mathbb{Z}
$

Fourier analysis tells us that the function space on $ S^1$ is topologically spanned by

% latex2html id marker 731
$\displaystyle \{f_k (x)=e^{\sqrt{-1} k x} ; k\in \mathbb{Z}\}.
$

Let us consider the Laplacian $ \Delta=-(\partial/\partial x)^2$ .

$\displaystyle \Delta( f_k(x))=k^2 f_k(x).
$

$\displaystyle \Delta= {\mathrm{diag}}(\dots,3^2,2^2,1^2,0^2,1^2,2^2,3^2,\dots)
$

The 0 -eigenspace of the Laplacian corresponds to the space of constants $ H^0(S^1,\mathbb{R})$ . We omit it and consider somewhat reduced matrix:

$\displaystyle \tilde \Delta= {\mathrm{diag}}(\dots,3^2,2^2,1^2,1^2,2^2,3^2,\dots).
$

We may then consider its $ -s/2$ -th power:

$\displaystyle \tilde \Delta^{-s/2}
= {\mathrm{diag}}(\dots,3^{-s},2^{-s},1^{-s},1^{-s},2^{-s},3^{-s},\dots).
$

Its trace is equal to the Riemann Zeta function (up to the multiplicative constant $ 2$ .)

$\displaystyle \operatorname{tr}\tilde \Delta^{-s/2}
=
2 \sum _{n=1}^\infty \frac{1}{n^s}= 2 \zeta(s).
$

REMARK 01.1   For any compact Riemannian manifold $ M$ , we may mimic the argument. Namely,
  1. There are Laplacians $ \Delta_i $ on the space $ \Omega^i(M,\mathbb{C})$ of $ i$ -forms for $ i=0,1,2,\dots, \dim M$ .
  2. For each $ i$ , the 0 -eigenspace of $ \Delta_i $ is equal to the $ i$ -th cohomology
  3. For each $ i$ , we may consider reduced Laplacian $ \tilde \Delta_i$ . It is an self adjoint operator with positive eigenvalues.
  4. The trace of the power $ \Delta_i ^{-s/2}$ may be considered as the ``$ i$ -th Zeta function of $ M$ ''.

REMARK 01.2   Eigen vectors of the Laplacians are ``particles'' on the manifold $ M$ . The zeta functions are then considered to be ``generating functions of the number of particles on $ M$ ''.

In general, we employ the following principle:

The zeta functions are ``generating functions of number of particles''

The meaning of the term ``particles'' may vary.


next up previous
Next: Formal power series Up: Zeta functions Previous: Zeta functions
2015-04-10