Difference between revisions of "Riemann Zeta function"
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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 | + | 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> | ||
Revision as of 01:04, September 25, 2008
The Riemann Zeta function, commonly denoted is a very important function in number theory. The function can be defined as 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 ). The function was investigated by a number of mathematicians including Euler who showed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(2) = \frac{\pi^2}{6}} but was first extensively investigated by Bernhard Riemann who showed that the behavior of the function was intimately connected to the distribution of 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 Riemann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}}
and then to all complex numbers via,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(1-s)=2^{1-s}\pi^{-s}\sin\left(\frac{1-s}{2}\pi\right)\Gamma(s-1)\zeta(s)} .
Two thing are noticible from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the real line 1/2.