Proof of The Euler Product Formula
This sketch of a proof only makes use of simple algebra commonly taught in high school. This was originally the method by which Euler discovered the formula. There is a certain sieving property that we can use to our advantage:
Subtracting the second from the first we remove all elements that have a factor of 2:
Repeating for the next term:
Subtracting again we get:
where all elements having a factor of 3 or 2 (or both) are removed.
It can be seen that the right side is being sieved. Repeating infinitely we get:
Dividing both sides by everything but the ζ(s) we obtain:
This can be written more concisely as an infinite product over all primes p:
To make this proof rigorous, we need only observe that when, the sieved right-hand side approaches 1, which follows immediately from the convergence of the Dirichlet series for ζ(z).
Read more about this topic: Proof Of The Euler Product Formula For The Riemann Zeta Function
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