Riemann Zeta Function
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Riemann Zeta Function |
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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 "analytic")
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.