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Many thanks,

Jameson

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- #1

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Many thanks,

Jameson

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shmoe

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I don't think I've looked at MathWorlds entry on Zeta before now. There's quite a bit there to try to take in at once if this is your first stab at it. You might want to have a look around for some simpler introductions like http://planetmath.org/encyclopedia/RiemannZetaFunction.html [Broken]. Also try searching this forum (specifically the number theory section) for "zeta" or "riemann", I don't know if there's anything like an introduction but most of what there is attempts to be basic (though that's difficult). Start with that and I'll be happy to try to answer any more specific questions you may have, otherwise I might be typing all night. There's also been several popsci books on the subject in recent years if you want really friendly introductions.

It's important because of it's connection to the primes. For real part of s>1 we have [itex]\zeta(s)=\sum_{n=1}^{\infty}1/n^s=\prod_{p}(1-1/p^s)^{-1}[/itex], where this product is taken over all primes p (the first equality is the definition of Zeta-this sum converges when real part of s >1). This is sometimes refered to as the analytic version of the fundamental theorem of arithmetic- think about expanding the product using [tex](1-1/p^s)^{-1}=1+1/p+1/p^2+1/p^3+\ldots[/tex]. The PlanetMath article mentions how this can show there are infinitely many primes.

The nontrivial zeros of Zeta (the ones with positive real part) are important because they influence the location of prime numbers (see the Prime Number Theorem). There's something usually called Riemann's explicit formula that makes this connection concrete by expressing the prime counting function as a certain sum involving the non-trivial zeros of Zeta. The Riemann Hypothesis is about the location of the non-trivial zeros, specifically that they all lie on the critical line. If we knew this were true, then we'd know more about the distribution of the primes. This has some real consequences for things like primality testing algorithms.

It's important because of it's connection to the primes. For real part of s>1 we have [itex]\zeta(s)=\sum_{n=1}^{\infty}1/n^s=\prod_{p}(1-1/p^s)^{-1}[/itex], where this product is taken over all primes p (the first equality is the definition of Zeta-this sum converges when real part of s >1). This is sometimes refered to as the analytic version of the fundamental theorem of arithmetic- think about expanding the product using [tex](1-1/p^s)^{-1}=1+1/p+1/p^2+1/p^3+\ldots[/tex]. The PlanetMath article mentions how this can show there are infinitely many primes.

The nontrivial zeros of Zeta (the ones with positive real part) are important because they influence the location of prime numbers (see the Prime Number Theorem). There's something usually called Riemann's explicit formula that makes this connection concrete by expressing the prime counting function as a certain sum involving the non-trivial zeros of Zeta. The Riemann Hypothesis is about the location of the non-trivial zeros, specifically that they all lie on the critical line. If we knew this were true, then we'd know more about the distribution of the primes. This has some real consequences for things like primality testing algorithms.

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mathwonk

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this is alittle simpler but gives an idea of the usefulness of such sums and such functions:

consider the sum:

Summation 1/p^s, summed over all primes. If we knew this function behaved like log(1/(s-1)) for s near 1, it would follow in particualr that it goes to infinity there, hence there are an infinite number of primes.

dirichlet showed that the analogous sum, but summed only over all primes congruent to a mod b, where a and b are relatively prime, also behaves like a constant times log(1/(s-1)); hence there are also infinitely many such primes.

this implies for instance that there are infinitely primes ending in each of 1,3,7 and 9.

in fact modifications of riemann zeta function callwed L functions of form

summation chi(n)/n^s summed over all integers n, where chi is a "character" [a extension of a complex valued homomorphism on the integers mod n] play a crucial role in the proof of this theorem.

consider the sum:

Summation 1/p^s, summed over all primes. If we knew this function behaved like log(1/(s-1)) for s near 1, it would follow in particualr that it goes to infinity there, hence there are an infinite number of primes.

dirichlet showed that the analogous sum, but summed only over all primes congruent to a mod b, where a and b are relatively prime, also behaves like a constant times log(1/(s-1)); hence there are also infinitely many such primes.

this implies for instance that there are infinitely primes ending in each of 1,3,7 and 9.

in fact modifications of riemann zeta function callwed L functions of form

summation chi(n)/n^s summed over all integers n, where chi is a "character" [a extension of a complex valued homomorphism on the integers mod n] play a crucial role in the proof of this theorem.

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Daniel.

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saltydog

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[tex]\zeta(z)=\frac{\Gamma(1-z)}{2\pi i}\oint\frac{u^{z-1}}{e^{-u}-1}du[/tex]

With I think the path going from [itex]-\infty[/itex] to [itex]-\infty[/itex]

Although I'm a bit confussed about that path. Maybe I have it wrong.

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shmoe

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[tex]\zeta(s)=\frac{e^{-i\pi s}\Gamma(1-s)}{2\pi i}\int\frac{u^{s-1}}{e^{u}-1}du[/tex]

where the contour of integration starts on the real axis at +infinity, comes down to the origin, circles once in the counterclockwise direction (avoiding the poles at +/-2*pi*i), then heads back to +infinity along the real axis. The branch of the logarithm (so we know what u^(s-1) is) varies from 0 to 2pi along this contour (the path 'back' to infinity is on a different branch then the path 'from' infinity).

Now this actually defines an analytic function on the entire plane (with a pole only at s=1 of course, the other poles of the Gamma factor are canceled, the integral is zero at these points). You can then go on to show this is equal to the usual Dirichlet series definition when real part of s is greater than 1 and you've therefore managed to find an analytic continuation of this series to the entire plane.

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saltydog

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Thanks a bunch Shmoe for that nice explanation. It's not immediately obvious to me but I'll make a hard-copy and pin it on my board to work on it at a later date. May need some help then.shmoe said:

[tex]\zeta(s)=\frac{e^{-i\pi s}\Gamma(1-s)}{2\pi i}\int\frac{u^{s-1}}{e^{u}-1}du[/tex]

where the contour of integration starts on the real axis at +infinity, comes down to the origin, circles once in the counterclockwise direction (avoiding the poles at +/-2*pi*i), then heads back to +infinity along the real axis. The branch of the logarithm (so we know what u^(s-1) is) varies from 0 to 2pi along this contour (the path 'back' to infinity is on a different branch then the path 'from' infinity).

Now this actually defines an analytic function on the entire plane (with a pole only at s=1 of course, the other poles of the Gamma factor are canceled, the integral is zero at these points). You can then go on to show this is equal to the usual Dirichlet series definition when real part of s is greater than 1 and you've therefore managed to find an analytic continuation of this series to the entire plane.

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