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
We know that the left-hand side of the equation diverges to infinity therefore the numerator on the right-hand side (the series of primes) must also be infinite for divergence.
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.
Patent Issued for Floating Point Multiplier with First and Second Partial Product Shifting Circuitry for Result Alignment
Jun 25, 2013; By a News Reporter-Staff News Editor at Information Technology Newsweekly -- ARM Limited (Cambridge, GB) has been issued patent...
US Patent Issued to ARM on June 11 for "Floating Point Multiplier with First and Second Partial Product Shifting Circuitry for Result Alignment" (Texas Inventor)
Jun 11, 2013; ALEXANDRIA, Va., June 11 -- United States Patent no. 8,463,834, issued on June 11, was assigned to ARM Ltd. (Cambridge, Great...
Patent Application Titled "Apparatus and Method for Generating Partial Product for Polynomial Operation" Published Online
Sep 19, 2013; By a News Reporter-Staff News Editor at Politics & Government Week -- According to news reporting originating from Washington,...