### III.47 *L*-Functions

* Kevin Buzzard*

### 1 How Can We “Package” a Sequence of Numbers?

Suppose we are given a sequence of numbers such as

How can we package up this sequence into *one* object that remembers everything about the sequence, and that might even give us new insights into the sequence? One standard technique is to use a GENERATING FUNCTION [III.32], but here is another way, which has proved very fruitful in number theory and elsewhere. Given a sequence *a*_{1}, *a*_{2}, *a*_{3}, . . . , we define the *Dirichlet series*

Here, *s* could be a positive integer, ...