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
asymptotic formula for \pi(x).
Maybe a simpler example of asymptotics is in order. (x^2+x)\sim x^2 means as x gets really large the ratio(x^2+x)/x^2 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 x^2 to approximate x^2+x when x is large. This approximation is imperfect though,especially if you want exact values of x^2+x. If you put x=10^{10}, then even though the ratio (x^2+x)/x^2 is within 10 decimal places of 1 the absolute error (x^2+x)-x^2 is an enormous 10^{10}. In terms of the graphs of the functions x^2+x and x^2 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 \pi(x) 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 \pi(x) 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 {\mbox \zeta(s)=\prod_{p\ \text{prime}}(1-p^{-s})^{-1}=\sum_{n=1}^{\infty}\frac{1}{n^{s}} (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 {\mbox\sum_{n=1}^{\infty}\frac{1}{n}} 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
complex magic (pun intended) to show there was a way to extend the definition of the zeta function to allow all complex values.
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 \pi(x) 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 \pi(x) was a grand thing. Up to this point, Gauss had conjectured {\mbox\pi(x)\sim \int_{2}^{x}\frac{1}{\log(t)}dt}, 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!)
