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:
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