## 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. [Read More]