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The '''Riemann Zeta function''', commonly denoted <math>\zeta (s)</math> is a very important function in number theory. The function can be defined as <math>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math> when s has a real part greater than 1.  The function can then be analytically continued for everywhere where s is non-one (the function has a simple pole at <math>s=1</math>). The function was investigated by a number of mathematicians including Euler who showed that <math>\zeta(2) = \frac{\pi^2}{6}</math> but was first extensively investigated by [[Bernhard Riemann]] who showed that the behavior of the function was intimately connected to the distribution of [[prime number|prime numbers]]. In particular, where the function is zero is related very closely to the error term of the [[Prime Number Theorem]] and the [[prime counting function]]. The [[Riemann hypothesis]] is an unsolved conjecture that describes strong restrictions on the locations of the zeros.  

The '''Riemann Zeta function''', commonly denoted <math>\zeta (s)</math> is a very important function in number theory. The function can be defined as <math>\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math> when s has a real part greater than 1.  The function can then be analytically continued for everywhere where s is non-one (the function has a simple pole at <math>s=1</math>). The function was investigated by a number of mathematicians including Euler who showed that <math>\zeta(2) = \frac{\pi^2}{6}</math> but was first extensively investigated by [[Bernhard Riemann]] who showed that the behavior of the function was intimately connected to the distribution of [[prime number|prime numbers]]. In particular, where the function is zero is related very closely to the error term of the [[Prime Number Theorem]] and the [[prime counting function]]. The [[Riemann hypothesis]] is an unsolved conjecture that describes strong restrictions on the locations of the zeros (specifically, that all non-trivial zeroes must lie on the line x=1/2).  

The convergence of the Reimann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,

The convergence of the Reimann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,