# 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