# Riemann zeta function

Personal notes on the Riemann zeta function and the Riemann hypothesis.

Notes taken while taking a course [1].

## Riemann zeta function

For $$s \in \Complex$$ with $$\mathrm{Re}(s) > 1$$,

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$

$$\sum_{n=1}^\infty \frac{1}{n^s}$$ converges for all $$\mathrm{Re}(s) > 1$$. The zeta function has an analytic continuation into $$\Complex - \{1\}$$.

All $$s$$ such that $$\zeta(s)=0$$ satisifies either $$s=-2n$$ for $$n>0$$ or $$0 < \mathrm{Re}(s) < 1$$.

## Riemann hypothesis

All $$s$$ such that $$\zeta(s)=0$$ and $$0 < \mathrm{Re}(s) < 1$$ satisfies $$\mathrm{Re}(s) = \frac{1}{2}$$.

## Relation to prime numbers

### Real zeta function

For $$s \in \mathbb{R}$$, thanks to unique prime factorization, it is the case that

$\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \sum_{k=0}^\infty \frac{1}{p^{ks}} = \prod_p \frac{1}{1-p^{-s}}$

### Prime number theorem

$$\pi(x)$$, the prime counting function whose value is the number of prime numbers less than or equal to $$x$$, is asymptotic to $$\frac{x}{\ln x}$$ as $$x \rightarrow \infty$$.

$\pi(x) \sim \frac{x}{\ln x}$

This is equivalent to

$\lim_{x \rightarrow \infty} \frac{\pi(x)}{\frac{x}{\ln x}} = 1$

### Tight error bound

The Riemann hypothesis is equivalent to

$\left| \pi(x) - \mathrm{li}(x) \right| < \frac{\sqrt{x} \ln x}{8\pi}$

where

$\mathrm{li}(x) = \int_0^x \frac{1}{\ln t} \, dt$