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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 [[analysis|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]]s. 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 '''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 [[analysis|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]]s. 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 number]]s first by stretching to converge over s with real part greater than zero,

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

<math>\zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}</math>

<math>\zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}</math>