ZETA FUNCTIONS

YOSHIFUMI TSUCHIMOTO

1.1. Formal power series.

Definition 1.1. 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.

1.2. 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.

1.2.1. Ordinary generating function.

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

Example 1.2. (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}} .

YOSHIFUMI TSUCHIMOTO

1.2.2. Dirichlet series generating function.

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

Proposition 1.3. (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}} ) .