Riemann Zeta Function


The Euler product formula for the Riemann Zeta Function reads

The result amazes me because its not very intuitive when presented initially

where the left hand side equals the Riemann zeta function:

and the product on the right hand side extends over all prime numbers p:

Proof :

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:


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