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 continued 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.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)