ZETA FUNCTIONS

YOSHIFUMI TSUCHIMOTO

1.1. Introduction.

S^{1} = ℝ/2πℤ

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

\{f_{k}(x) = e^{\sqrt{−1}kx}; k ∈ ℤ\}.

Let us consider the Laplacian Δ = −(∂/∂x)^{2}.

Δ(f_{k}(x)) = k^{2}f_{k}(x).
Δ = diag(…, 3^{2}, 2^{2}, 1^{2}, 0^{2}, 1^{2}, 2^{2}, 3^{2}, …)

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

Δ̃ = diag(…, 3^{2}, 2^{2}, 1^{2}, 1^{2}, 2^{2}, 3^{2}, …).

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

Δ̃^{−s/2} = diag(…, 3^{−s}, 2^{−s}, 1^{−s}, 1^{−s}, 2^{−s}, 3^{−s}, …).

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

tr Δ̃^{−s/2} = 2 ∑_{n=1}^{∞} \frac{1}{n^{s}} = 2ζ(s).

Remark 1.1. For any compact Riemannian manifold M, we may mimic the argument. Namely,

(1) There are Laplacians Δ_{i} on the space Ω^{i}(M, ℂ) of i-forms for i = 0, 1, 2, …, dim M.

(2) For each i, the 0-eigenspace of Δ_{i} is equal to the i-th cohomology (3) For each i, we may consider reduced Laplacian Δ̃_{i}. It is an self

adjoint operator with positive eigenvalues.

(4) The trace of the power Δ_{i}^{−s/2} may be considered as the “i-th Zeta function of M”.

Remark 1.2. 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:

[figure]

The zeta functions are “generating functions of number of particles” The meaning of the term “particles” may vary.

YOSHIFUMI TSUCHIMOTO

1.2. Formal power series.

Definition 1.3. Let A be a commutative ring. Let X be a variable. A formal power series in X over A is a formal sum

∑_{i=0}^{∞} a_{i}X^{i} (a_{i} ∈ A)

We denote by A[[X]] the ring of formal power series in X. Namely,

A[[X]] = {∑_{i=0}^{∞} a_{i}X^{i}; a_{i} ∈ A}.

For any element f = ∑_{n} a_{n}X^{n} of A[[X]], we define its order as follows:

ord(f) = inf{n; a_{n} ≠ 0}.

Then we may define a metric on A[[X]].

d(f, g) = \frac{1}{2^{ord(f−g)}}

Exercise 1.1. Show that (A[[X]], d) is a complete metric space.

Exercise 1.2. Show that A[[X]] is a topological ring. That means, it is a topological space equipped with a ring structure and operations (the addition and the multiplication) is continuous.

1.3. generating functions. A generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers {a_{n}} that is indexed by the natural numbers.

1.3.1. Ordinary generating function.

G_{0}({a_{n}}; X) = ∑_{n=0}^{∞} a_{n}X^{n}

Example 1.4. (Examples of ordinary generating functions)

(1) A generating function of a geometric progression:

∑_{i=0}^{n} a^{n}X^{n} = \frac{1}{1 − aX} .

(2) A generating function of an arithmetic progression:

∑_{i=0}^{n}(n + 1)X^{n} = \frac{1}{(1 − X)^{2}} .

1.3.2. Dirichlet series generating function.

G_{1}(\{a_{n}\}; s) = ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}

Proposition 1.5. (Euler product expression) Assume {a_{n}} is multiplicative in the sense that

gcd(n, m) = 1 ⟹ a_{n}a_{m} = a_{nm}

holds, Then we have

∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = ∏_{p;prime} (∑_{k=0}^{∞} \frac{a_{p^{k}}}{p^{ks}} ) .