User:Peter Driscoll

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

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


 * where the product of sets $\prod_{a \in A} B_a$ is taken to be a cartesian product.

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.
 * $ h(v) = \left( \prod_{p \in P}^{p \le A} p^{v_p} \right) $

Define Q by,

Then,

From the Fundamental Theorem of Arithmetic,