Riemann zeta

Consider the function $\zeta$ defined by the series $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$

Properties

Proposition: If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges.

Proof: █

Proposition (Euler Product): $\zeta(z)=\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^z} = \displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}}$

Proof: █

External links

• 15 Videos about the Riemann $\zeta$ function

English translation of Riemann's paper "On the number of prime numbers less than a given quantity"