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Quick description
In order to evaluate some mathematical expression it is often benefitial to combine terms with a common property.
Prerequisites
Some real analysis.
Example 1
If one has the Riemann zeta function

then the sum is absolutely convergent for Re.
Therefore it can be rearranged in every possible manner. Now let

where means that
divides
but
does not.
Then the set
form a partition of
and we get

But now we can write as
where
does not divide
and we get

by the geometric sum formula.
Applying this to all the other primes and using a limiting argument we establish
the product formula
