Most of us are aware of the following consequence of Fundamental Theorem of Arithmetic:

There are infinitely many prime numbers.

The classic proof by Euclid is easy to follow. But I wanted to share the following two analytic equivalents (infinite series and infinite products) of the above purely arithmetical statement:

- diverges.

For proof, refer to this discussion: https://math.stackexchange.com/q/361308/214604

- , where is any complex number with .

The outline of proof, when is a real number, has been discussed here: http://mathworld.wolfram.com/EulerProduct.html