# Dirichlet's Theorem: Statement

![](attachments/Pasted%20image%2020220228153916.png)

## Strong Form and Proof
![](attachments/Pasted%20image%2020220228153933.png)
![](attachments/Pasted%20image%2020220228154015.png)

# Chebotarev Density
![](attachments/Pasted%20image%2020220228154024.png)
![](attachments/Pasted%20image%2020220228154153.png)
![](attachments/Pasted%20image%2020220228154324.png)
![](attachments/Pasted%20image%2020220228154504.png)
## Splitting of Primes

![](attachments/Pasted%20image%2020220228154243.png)![](attachments/Pasted%20image%2020220228154601.png)

# S&S

## The Group of Units

![](attachments/Pasted%20image%2020220228154755.png)

## Infinitude of Primes

![](attachments/Pasted%20image%2020220228154818.png)
![](attachments/Pasted%20image%2020220228154913.png)

## Product Formula

![](attachments/Pasted%20image%2020220228155236.png)

## Proof of Dirichlet

![](attachments/Pasted%20image%2020220228155439.png)

## Counting Primes in 4k+1

![](attachments/Pasted%20image%2020220228155521.png)
![](attachments/Pasted%20image%2020220228155541.png)

## Computing an L function

![](attachments/Pasted%20image%2020220228155645.png)

## Proof

![](attachments/Pasted%20image%2020220228155703.png)

Steps:

- Prove a product formula
- Make sense of taking log of both sides
- Third:
	![](attachments/Pasted%20image%2020220228160058.png)

### Fourier Expansion

![](attachments/Pasted%20image%2020220228155811.png)

## Logs of L functions

![](attachments/Pasted%20image%2020220228161359.png)

# Fourier Series

## Wave and Heat Equations

![](attachments/Pasted%20image%2020220228161854.png)
![](attachments/Pasted%20image%2020220228161904.png)

![](attachments/Pasted%20image%2020220228162053.png)


![](attachments/Pasted%20image%2020220228162144.png)

## Fourier Expansion
![](attachments/Pasted%20image%2020220228161953.png)
![](attachments/Pasted%20image%2020220228162302.png)
![](attachments/Pasted%20image%2020220228162317.png)
![](attachments/Pasted%20image%2020220228162336.png)
![](attachments/Pasted%20image%2020220228162408.png)
![](attachments/Pasted%20image%2020220228162428.png)
![](attachments/Pasted%20image%2020220228162516.png)
![](attachments/Pasted%20image%2020220228162525.png)
![](attachments/Pasted%20image%2020220228162554.png)

![](attachments/Pasted%20image%2020220228162652.png)
![](attachments/Pasted%20image%2020220228162717.png)
![](attachments/Pasted%20image%2020220228162729.png)