Perfect Symmetry

So far, we have seen how the Euler product links the \zeta-function to the prime numbers. More precisely, it encodes the fundamental theorem of arithmetic. One may also say, it's the analytic version of it, in a sense that should become clearer shortly.

What we have done so far works perfectly for the real numbers. The sum \sum n^{-s} that defines \zeta(s) converges for s>1, that's how Leonhard Euler found his product, and that's what Peter Gustav Lejeune Dirichlet used to prove the prime number theorem in arithmetic progressions. The ingenious step In Riemann's 1859 paper was to allow for complex values s. As mentioned before, the same argument as in the real case proves that the sum converges for \Re s>1. Since the convergence is absolute, \zeta(s) is analytic in this domain. If you don't know what analytic is, just read it as well-behaved or, even better, cool.

The shame is that -- as you certainly have already heard if you still read this blog -- all the action takes place in the critical strip, i.e. the area just to the left of our line of convergence with 0\le\Re s\le1. Riemann showed that we can calculate values of \zeta(s) for \Re s\le1 through the beautiful functional equation

\zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s).

Continue reading Perfect Symmetry

Does the Euler Product Converge?

Usually, I don't care too much about convergence as a general overview of the argument is what I aim at here, and otherwise I'll just trust that things "behave well". But some words concerning convergence won't harm.

It's a well known fact that the harmonic series (which we shortly touched in the previous post) \sum1/n diverges. I think the best (though not easiest) proof of this to compare it with the corresponding integral:

\sum_{n=1}^x\frac{1}{n}>\int_1^x\frac{1}{t}\mathrm{d}t=\log x\longrightarrow\infty.

(Let's pause for a moment to celebrate the first of the numerous appearances of our good friend the logarithm.) Continue reading Does the Euler Product Converge?

Euler Product Revisited

From the previous post we know that the harmless looking series \sum n^{-s} can be extended to the product \prod (1-p^{-s})^{-1}. At first sight, this does not seem terribly helpful, and it actually makes the rather easy series more complicated. So what's the big deal?

It's what has actually been suppressed in the above notation: The sequences we run through. The series runs over all natural numbers (or positive integers, if you prefer), the product runs through all prime numbers. Now, that's cool, isn't it? We found a series over the natural numbers that, as we will see later, defines a well-behaved function which is accessible to all the nice methods modern mathematics can offer, and related it to the prime numbers.

In other words: Riemann's \zeta-function encodes the mysteries of the primes. Continue reading Euler Product Revisited