where the left hand side equals the Riemann zeta function:
and the product on the right hand side extends over all prime numbersp:
Proof of the Euler product formula
The method of Eratosthenes used to sieve out prime numbers is employed in this proof.
This sketch of a proof only makes use of simple algebra that most high school students can understand. 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 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 Re(s) > 1, the sieved right-hand side approaches 1, which follows immediately from the convergence of the Dirichlet series for .
An interesting result can be found for
which can also be written as,
which is,
as,
thus,
We know that the left-hand side of the equation diverges to infinity therefore the numerator on the right-hand side (the primorial) must also be infinite for divergence.
Another proof
Each factor (for a given prime p) in the product above can be expanded to a geometric series consisting of the reciprocal of p raised to multiples of s, as follows
When , and this series converges absolutely. Hence we may take a finite number of factors, multiply them together, and rearrange terms. Taking all the primes p up to some prime number limit q, we have
where σ is the real part of s. By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ(s) goes to zero when σ > 1, we have convergence in this region.
references
John Derbyshire, Bernard Riemann and The Greatest Unsolved Problem in Mathematics, Joseph Henrry Press, 2003, ISBN 9780309085496