# Book:Article/Bernhard Riemann/Ueber die Anzahl der Primzahlen under einer gegebenen Grösse

## Journal Article

$\text {1859}$: Ueber die Anzahl der Primzahlen under einer gegebenen Grösse
Bernhard Riemann

## Subject Matter

In this thesis, Bernhard Riemann investigates Euler's definition of the Zeta function:

$\ds \sum_{n \mathop \ge 1} \frac 1 {n^s} = \prod_p \dfrac 1 {1 - p^{-s} }$

where the product ranges over all primes $p$.

Riemann recognised that the deeper questions about distribution of primes can be investigated only by $s$ being a complex variable.

## Historical Note

This was the only paper written by Bernhard Riemann on the subject of number theory.

It contains the statement of what is now known as the Riemann Hypothesis.