User:Peter Driscoll

Product over a sum is the sum over the cartesian products of the products

Define P as,

Change the summing variable using:

The Fundamental Theorem of Arithmetic, guarantees a unique factorization for each positive natural number. Therefore this function is one to one.
 * $\displaystyle h(v) = \prod_{p \in P}^{p \le A} p^{v_p} $

The change of variable gives,

Define Q by,

Then,

Consider,

The construction defines it as the set of all possible products of positive powers of primes. From the definition of a prime number, every positive natural number may be expressed as a prime or a product of powers of primes. Then,
 * $\displaystyle k \in \mathbb{N} \implies k \in W $

also every element of W is a positive natural number
 * $\displaystyle k \in W \implies k \in \mathbb{N} $

So $ W = \mathbb{N} $.

Then taking limits on P: