Difference between revisions of "Riemann zeta function"
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− | The '''zeta-function''' is a function very important to the [[Riemann Hypothesis]]. The function is <math>\zeta (s)= | + | The '''zeta-function''' is a function very important to the [[Riemann Hypothesis]]. The function is <math>\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots</math> |
The series is convergent [[iff]] <math>\Re(s)>1</math>. [[Leonhard Euler]] showed that when <math>x=2</math>, the sum is equal to <math>\frac{\pi^2}{6}</math>. Euler also found that since every number is the product of a certain combination of [[prime number]]s, the zeta-function can also be expressed as <math>{\zeta}(s) = \left(\frac{1}{(2^0)^s}+\frac{1}{(2^1)^s}+\frac{1}{(2^2)^s}+\cdots\right) \left(\frac{1}{(3^0)^s}+\frac{1}{(3^1)^s}+\frac{1}{(3^2)^s}+\cdots\right) \left(\frac{1}{(5^0)^s}+\frac{1}{(5^1)^s}+\frac{1}{(5^2)^s}+\cdots\right) \cdots </math>. By summing up each of these [[geometric series]] in parentheses, we have the following identity, the so-called [[Euler Product]]: <math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p\ \mathrm{prime}} (1-p^{-s})^{-1}</math>. | The series is convergent [[iff]] <math>\Re(s)>1</math>. [[Leonhard Euler]] showed that when <math>x=2</math>, the sum is equal to <math>\frac{\pi^2}{6}</math>. Euler also found that since every number is the product of a certain combination of [[prime number]]s, the zeta-function can also be expressed as <math>{\zeta}(s) = \left(\frac{1}{(2^0)^s}+\frac{1}{(2^1)^s}+\frac{1}{(2^2)^s}+\cdots\right) \left(\frac{1}{(3^0)^s}+\frac{1}{(3^1)^s}+\frac{1}{(3^2)^s}+\cdots\right) \left(\frac{1}{(5^0)^s}+\frac{1}{(5^1)^s}+\frac{1}{(5^2)^s}+\cdots\right) \cdots </math>. By summing up each of these [[geometric series]] in parentheses, we have the following identity, the so-called [[Euler Product]]: <math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p\ \mathrm{prime}} (1-p^{-s})^{-1}</math>. | ||
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The next step is the [[functional equation for the zeta function|functional equation]]: Let <math>\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)</math>. Then <math>\xi(s)=\xi(1-s)</math>. This gives us a meromorphic continuation of <math>\zeta(s)</math> to all of <math>\mathbb{C}</math>. | The next step is the [[functional equation for the zeta function|functional equation]]: Let <math>\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)</math>. Then <math>\xi(s)=\xi(1-s)</math>. This gives us a meromorphic continuation of <math>\zeta(s)</math> to all of <math>\mathbb{C}</math>. | ||
+ | {{wikify}} | ||
+ | [[Category:Number theory]] |
Revision as of 21:11, 15 November 2007
The zeta-function is a function very important to the Riemann Hypothesis. The function is
The series is convergent iff
. Leonhard Euler showed that when
, the sum is equal to
. Euler also found that since every number is the product of a certain combination of prime numbers, the zeta-function can also be expressed as
. By summing up each of these geometric series in parentheses, we have the following identity, the so-called Euler Product:
.
However, the most important properties of the zeta function are based on the fact that it extends to a meromorphic function on the full complex plane which is holomorphic except at , where there is a simple pole of residue 1. Let us see how this is done: First, we wish to extend
to
. To do this, we introduce the alternating zeta function
, which is convergent on
. (This follows from one of the standard convergence tests for alternating series.) We then have
. We therefore have
when
.
The next step is the functional equation: Let . Then
. This gives us a meromorphic continuation of
to all of
.
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