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- Thread starter Canute
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shmoe

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shmoe

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However, it is very much connected with prime numbers. In fact if real part of s>1, [tex]{\mbox \zeta(s)=\prod(1-p^{-s})^{-1}}[/tex], where this is a product over all primes p. The location af the zeros are also very related to the distribution of the primes, there's a formula for the prime counting function that involves an infinite sum over the non-trivial zeros of Zeta. As you can imagine, the location of these zeros influences the prime counting function.

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What I'm trying to understand is not the mathematics, but the principles involved. One question might be, if Reimann's hypothesis were proved then what would this allow us to say about the primes?

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shmoe

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Let [tex]\pi(x)=[/tex] number of primes less than or equal to x. so [tex]\pi(1)=0,\pi(4)=2,\pi(7.675)=4[/tex].

Then it's been shown that as x gets large

[tex]\pi(x)\sim \int_{2}^{x}\frac{1}{\log(t)}dt+junk[/tex]

This is nice, except for the mysterious 'junk' term. In a way the 'junk' measures how badly the primes are distributed compared to the very nice (trust me it's nice) integral part.The tough part of this junk term involves a nasty looking infinite sum involving the zeros of the zeta function (and also depending on x). If you could evaluate this sum precisely then you would know everything there is to know about the distribution of the primes.

Unfortunately this isn't possible since we don't know even know what all the zeros are. As a result, we settle on worst case scenario possibilities and try to get reasonable bounds on the 'junk' term.

The more we know about where the zeros live, the more we can say about the junk. If Riemann's hypothesis were true, then we'd know that the junk term is of the order of magnitude [tex]x^{1/2}\log^2(x)[/tex], this is good since as x grows it would mean our main term dwarfs the junk term (our main integral term is of the order of magnitude [tex]x/\log(x)[/tex]). It turns out the converse is true, if we know that's the order of magnitude of 'junk' then the Riemann hypothesis is also true. Before you ask, this actually seems like a very very bad way to try to prove the Riemann Hypothesis. In this way the location of the zeros and the distribution of the primes are tightly linked.

Our current best known bounds for junk come from the current best know zero free regions of zeta (they are a little nasty looking). It always been this way, as the zero free region is enlarged slightly, we know for certain that the primes are better behaved.

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Sorry for the naive questions but I'm trying to understand what it is that Reimann did with the Zeta function, or what it is that the function does, but without much (any?) idea of the actual mathematics involved.

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shmoe

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Canute said:Would it be correct to say that knowing the position of the zero's allows the number of primes up to x to be calculated more accurately than otherwise (than Gauss's calc for instance?).

That would be fair, but maybe not quite in the sense of the word "calculate" that you would mean. Better control of the zeros gives better bounds for the junk term in the

Maybe a simpler example of asymptotics is in order. [tex](x^2+x)\sim x^2[/tex] means as x gets really large the ratio[tex](x^2+x)/x^2[/tex] tends to 1. If you've taken any calculus, you've seen this plenty of times. If not, try putting in very large numbers to get an idea of what is happening. So we can use the simpler [tex]x^2[/tex] to approximate [tex]x^2+x[/tex] when x is large. This approximation is imperfect though,especially if you want exact values of [tex]x^2+x[/tex]. If you put [tex]x=10^{10}[/tex], then even though the ratio [tex] (x^2+x)/x^2[/tex] is within 10 decimal places of 1 the absolute error [tex](x^2+x)-x^2[/tex] is an enormous [tex]10^{10}[/tex]. In terms of the graphs of the functions [tex]x^2+x[/tex] and [tex]x^2[/tex] if you zoomed out very far, the graphs would be indistinguishible, but up close there's a huge gap.

Now control on the junk term in the prime number theorem tells us how fast the ratio of [tex]\pi(x)[/tex] and our simpler formula (such as Gauss's logarithmic integral estimate I gave in the last post) is going to 1. The absolute error can (and does) still get extremely large. This means we will never be able to calculate [tex]\pi(x)[/tex] to the nearest integer using the prime number theorem.

However, asymptotics are good for many applications. For example, having a better control of the junk will let us calculate the distribution well enough to say certain primality testing algorithms work properly. I haven't really given you a sense of what I mean by calculate here, but hopefully I've given you more sense of what it's not.

Canute said:Sorry for the naive questions but I'm trying to understand what it is that Reimann did with the Zeta function, or what it is that the function does, but without much (any?) idea of the actual mathematics involved.

That groovy expression for zeta I gave a couple posts back has a form in terms of an infinite sum [tex]{\mbox \zeta(s)=\prod_{p\ \text{prime}}(1-p^{-s})^{-1}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}[/tex] (by the way, this second equality can be thought of as an analytic representation of the fundamental theorem of arithmetic). You may recognize this better, if s=1 you get the harmonic series [tex]{\mbox\sum_{n=1}^{\infty}\frac{1}{n}}[/tex] which you may have seen diverges to infinity.

Before Riemann, Euler had considered this infinite sum only for real values of s. Riemann allowed s to wander over the complex plane. A problem was the infinite sum (or the infinite product over the primes) was not well behaved if the real part of s was less than or equal to 1 (this is directly related to the divergence of the harmonic series above). Riemann used some

Riemann then went on to do many great things. He showed that the zeta function had no zeros with real part greater than 1. He conjectured (possibly had a proof for) very accurate estimates on the number of zeros in the critical strip. He proved a formula that gives [tex]\pi(x)[/tex] explicitly in terms of the zeros of zeta. This main term in this formula was also more accurate that Gauss's, though he was unable to prove that it was in fact the 'main term' (meaning the junk was small). And of course he conjectured his famous hyposthesis.

His formula for [tex]\pi(x)[/tex] was a grand thing. Up to this point, Gauss had conjectured [tex]{\mbox\pi(x)\sim \int_{2}^{x}\frac{1}{\log(t)}dt}[/tex], but no one could prove it. Riemann's formula reduced this prolem to proving that the zeros were in 'the right locations'. In fact it turned out that if you could show there were no zeros on the line real part of s=1 then the junk term in Riemann's formula would be 'small enough' to conclude that Gauss's asymptotic estimate was correct. This was done, but not by Riemann. He laid out the tools needed to prove the prime number theorem.

I hope that gives you at least a very coarse outline of what's what. With the recent publicity of the clay prize ($1 million for solving the Riemann Hypothesis) there has been a few books aimed at a general audience on the subject, you might consider picking one up (if you're paying for it, look at it very closely to see if it has a level of math you're comfortable with). They'd probably do a better job of explaining things to you (though I'm happy to answer any questions you have!)

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Btw my interest stems from the resonances between some of the functions relating to primes and those related to the behaviour of the harmonic series and vibrating strings. It's damn annoying not being able to understand the mathematics. Thanks for the help anyway.

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shmoe

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It's a shame that some of the most beautiful objects in mathematics are very difficult to explain to the general public.

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kreil

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Canute, unfortunately the Riemman conjecture is far too advanced to be lowered to any level below basic calculus, and still retain any amount of meaning. I have also read de Sautoy's music of the primes, and I think it is one of the best books on the subject for laymen. I have read 2 or 3 other books on the same subject...one of which reduced it a little too much (the concepts were so simplified they got in the way of me understanding what I had previously read) and the other of which was beyond my grasp. Being only a highschool student currently taking calculus, I am in the same boat as you. I recommend you do what I did, just read Music of the Primes over and over and work on the math involved until you master the concepts and get a good handle on the math, then move on to a more challenging book. In this manner you can work your way up.

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