Tricki

## Order by property

### 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