shmoe

There's an old joke about this problem (I modified the answer to match the version at hand): ---- When this problem was posed to John von Neumann, he immediately replied, "90 miles." "It is very strange," said the poser, "but nearly everyone tries to sum the infinite series." "What do you mean, strange?" asked Von Neumann. "That's how I did it!" ---- The infinite series mentioned is the 'obvious' way of setting up the problem and computationally more work. Each term in the series is one leg of the flys journey. If you hope to teach a method like this to your classmates I'd suggest you try to set this sum up yourself, it's the best way to learn it before you attempt to teach it.

Does your desire to not pay outweigh this belief? It's stealing, there's no two ways about it. You're getting a product you didn't pay for. If you're ok with that, go ahead and download. Otherwise pony up the cash or go without. Is there any way you can log into your department remotely and access matlab that way? (This is possible at my department)

Do you mean: [math]\frac{\pi^2}{6}=\sum_{n=1}^\infty n^{-2}=\prod_{p\ \text{prime}}(1-p^{-2})^{-1}[/math] Euler argued the first equality by comparing coefficients of the taylor series of sin(x) with it's product form (which hadn't been fully justified until Hadamard). Here's a bunch of different ways to prove it (Euler's is #7): http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf The second equality above is just a special case of the Euler product form for the Riemann Zeta function (<-words to punch into google). It requires unique factorization and some arguments about convergence.

Ahh, good. k=0 should be familiar as well.

You've shown [math]\sigma_k[/math] is mutiplicative so it's enough to evaluate it on prime powers, which is just a geometric series.

At the nth step, the mth bulb is flipped if and only if n divides m, so you want to determine which m's have an odd number of divisors. Consider the prime factorization of m etc..

The prime number theorem asserts that [math]\pi(x)\approx Li(x)[/math], where [math]Li(x)=\int_{2}^{x}\frac{dt}{\log t}[/math]. The question of just how good this approximation is is directly related to the location of the zeros- roughly the smaller the error the closer RH is to being true. If you can show that [math]\pi(x)=Li(x)+O(\sqrt{x}\log x)[/math] by some method then you'll have proven the Riemann hypothesis (conversely RH implies this error term). There are elementary proofs of the prime number theorem (e.g. Selberg and Erdos), "elementary" meaning it avoids analytic methods and appealing to results about Zeta, not that they are simple. The error terms that these proofs are able to show will in turn say something about the zeros of zeta. However, they imply much weaker results about the zeros than we can achieve by attacking them directly with analytic techniques. You also might want to look up the Merten's function. It's pretty simple to understand and has a nice related equivalence with RH. (though this also seems like a hopeless way to tackle RH) Nahh, it's a very, very complicated beast. If you haven't already, you might want to check out some of the "popsci" books on Zeta. My usual reccomendation is to start with Derbyshire's "Prime Obsession" and work your way up to Edward's textbook (which also contains a translation of Riemann's original paper). Check out http://www.math.ubc.ca/~pugh/Psi/ it's a nice version of what I called a "good exercise" in my last post. There's some other nice applets on this site as well. It has the explicit formula at the top (not for pi(x) but a related function) that you should be able to understand the bits of if you understand complex exponentials (which aren't that difficult) and the fact that the zeros come in conjugate pairs.

"The question about waves came from reading that the zeros are in some way like tuning forks, and that the interaction of their different frequencies in some way defines the primes. Is this roughly correct?" Yes. You can think of the frequency being determined by the imaginary part of the zero, and the amplitude being determined by the real part of the zero. The furthur the zero is from the critical line (real part equals 1/2) the larger the amplitude, and in some sense the worse behaved the corresponding "wave" is (it's like you have one really loud tuning fork), and therefore the more erratic the distribution of the primes (worse error term in the prime number theorem). A good exercise is to take the Riemann-von Mangoldt explicit formula and the first few hundred zeros (ordered by increasing imaginary part, zeros can be found on Andrew Odlyzko's website for example) and see how close the formula comes to giving the prime counting function with the truncated sum over your finite number of zeros (vary the number of zeros as well). You can also plot the "wave" each zero corresponds too. To prove RH, you don't have to know exactly the locations of the zeros, 'just' that their real part is 1/2. It's not necessary to predict their imaginary parts (though quite alot is known about this already, like asymptotics for the number of zeros less than a certain height). I'm not sure exactly what you have in mind for "predict(ing) the position of the primes to infinity", but a way to quantify this would be bounds on the error term in the prime number theorem. In this interpretation, you could prove RH by knowing enough about the position of the primes (like I said before though, this is not a promising method to prove RH). If you want to quantify all of this handwaving about how zeros and primes are related, you should look up the explicit formula. There's plenty of online resources to be found, and plenty of good texts (e.g. Edwards, Ivic).

Well the zeros and primes are already known to be intimately linked via the Riemann-von Mangoldt explicit formula (stated by Riemann, proven by von Mangoldt). It gives the prime counting function (the number of primes up to x) in terms of an infinite sum indexed by the nontrivial zeros of zeta. Each term in this sum is a kind of wave like I think you're after. If you've managed to somehow prove a zero free region for zeta (in the critical strip) you can use this explicit formla to translate to a bound for the error term in the prime number theorem, and vice versa (!). So theoretically you could prove the Riemann hypothesis by proving the error term in the PNT is "small enough". However this would be unlikely- historically the strongest error terms have come directly from the strongest zero free regions, not the other way around. It would also be very odd if that ended up happening, after all the zeta function is studied to give insight into primes.

You want an N so that: [math]\sum_{n=1}^{\infty}\frac{n^2}{3^n}-P_N<10^{-6}[/math] where [math]P_N=\sum_{n=1}^{N}\frac{n^2}{3^n}[/math] (there's no dependance on "x" here). The left side is just a sum from N+1 to infinity. Use their hint and dominate this sum with a geometric series to get an upper bound dependant on N. Choose N appropriately.