Consider the following infinte sequence:

$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$

This is convergent only for values of s whose real part is greater than 1. However, there is a unique way of extending it to the entire complex plane (except at s=1 ) such that the result is still "nice" (for which read "is an analytic function")

This is then a function of one complex variable, and is named after Bernhard Riemann.

Leonhard Euler showed the following identity is true:

$\begin{matrix}\sum_{n=1}^\infty\frac{1}{n^s}&=&\prod_{p\text{~prime}}&1/(1-p^{-s})\end{matrix}$

This creates a connection between the Riemann Zeta function and the prime numbers.

There is much that is unknown about the Riemann Zeta function. For example, it's not known exactly where it takes the value 0. The Riemann Hypothesis is related to this question.

NamingPhilosophy
Root
(none) (none)
RiemannHypothesis (none)

## You are here

RiemannZetaFunction
BernhardRiemann
ComplexPlane
Euler
Function
Logarithm RealNumber ArgandDiagram
Calculus
CoDomainOfAFunction
ComplexNumber
CompositeNumber
CountingNumber
Divisor
DomainOfAFunction
Euclid
FamousPeople
GraphTheory
ImageOfAFunction
ImaginaryNumber
Polynomial
PrimePair
RangeOfAFunction
WholeNumber