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 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]]. 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. |
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| + | 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, | ||
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| + | <math>\zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}</math> | ||
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| + | and then to all complex numbers via, | ||
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| + | <math>\zeta(1-s)=s^{1-s}\pi^{-s}sin\left(\frac{1-2}{2}\pi\right)\Gamma(s-1)\zeta(s)</math>. | ||
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| + | 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 1/2 line. | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||
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Revision as of 23:48, May 3, 2008
The Riemann Zeta function, commonly denoted is a very important function in number theory. The function can be defined as 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) = \sum_{n=1}^\infty \frac{1}{n^s} } 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 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 s=1} ). 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.
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,
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)=s^{1-s}\pi^{-s}sin\left(\frac{1-2}{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 1/2 line.