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Sovemp
Tufted Titmouse
Joined: Jan 16, 2008
Age: 29
Posts: 27
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 Posted: Mon Jun 09, 2008 12:21 pm Post subject: Question about the Riemann zeta function |
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Hello all. So I'm basing http://en.wikipedia.org/wiki/Riemann_zeta_function for most of my understanding of this.
The Wikipedia article states that the Riemann Zeta function Z(s) is a function of a complex variable s initially defined by the infinite series:
for values of s with real part greater than 1.
So, in a listing of common values, for example it says Z(1) = 1 + 1/2 + 1/3 + ..., the harmonic series. Fine, cool, pretty straightforward.
However, it says that Z(0) = -1/2. Now, how is this? Obviously this doesn't seem to work for the initial equation I posed above, as by that Z(0) would be 1 + 1 + 1 + ... I'm assuming that Z(0) is defined differently since it states the initial equation is valid for only s > 1 in the real numbers. So, can anyone explain to me how this function is calculated for z = 0?
Thanks.
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lau
Really nice person to know. :)
Joined: Jun 18, 2006
Age: 60
Posts: 9482
Location: Somerset UK
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| Posted: Mon Jun 09, 2008 1:59 pm Post subject: |
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I think you just stopped reading the Wikipedia article too early...
...and then analytically continued to all complex s ≠ 1.
See this bit:
http://en.wikipedia.org/wiki/Riemann_zeta_function#Hadamard_product
Other than the infinite pole at s=1, it is a well behaved function.
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wolphin
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Joined: Aug 16, 2007
Posts: 502
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| Posted: Mon Jun 09, 2008 11:39 pm Post subject: |
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There are a couple alternative ways to do it.
The "modern" way to do it is to consider the infinite series as giving the values of the zeta function on a subset of the complex plane - namely the part of the complex plane with real part greater than 1. In this domain, the infinite series defines a so-called "analytic function" - meaning it can be expanded in a taylor series around any point s with Re(s) > 1.
Now, it's a property of analytic functions that you can perform a trick called analytic continuation - you can take those taylor series (whose radius of convergence will always be the distance to the nearest singularity) and expand those series around points far away from the singularity at s = 1. The region of convergence of these series, crucially, will include points with real part less than or equal to 1, which now defines values on those regions - such values are called a direct analytic continuation of the original function. (http://en.wikipedia.org/wiki/Analytic_continuation)
With the function defined by the zeta series, you can use repeated applications of direct analytic continuation to so-called "analytically continue" to the rest of the complex plane (that is, to find taylor series at every point), except for at s=1. It is this "analytic continuation" that most people refer to as the "riemann zeta function".
There are other methods to get it, such as riemann's original method (which unfortunately I'm not that familiar with) as well as non-cauchy summation methods, which are alternative ways to define sums of infinite series (i.e. http://en.wikipedia.org/wiki/Ramanujan_summation) |
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Sovemp
Tufted Titmouse
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Age: 29
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| Posted: Mon Jun 16, 2008 5:59 pm Post subject: |
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Thanks for the replies. That makes a lot more sense now.
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