Order by property

This means that it cannot be considered to contain or lead to any mathematically interesting information.

In order to evaluate some mathematical expression it is often a benefit to combine terms with a common property.

Some real analysis.

If one has the Riemann zeta function

$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$

then the sum is absolutely convergent for Re $(s) > 1$ .

Therefore it can be rearranged in every possible manner. Now let

$2^h \|\ n \}$

where $2^h \|\ n$ means that $2^h$ divides $n$ but $2^{h+1}$ does not. Then the $A_h, h = 0,1,2...$ form a partition of $\mathbb{N}$ and we get

$\zeta(s) = \sum_{h = 0}^\infty\sum_{n \in A_h} \frac{1}{n^s}$

But now we can write $n^s$ as $2^{hs}m^s$ where $2$ does not divide $m$ and we get

$\zeta(s) = \sum_{h = 0}^\infty \frac{1}{2^{hs}}\sum_{m \in A_0}\frac{1}{m^s} = \left(1 - \frac{1}{2^s}\right)^{-1}\sum_{m \in A_0}\frac{1}{m^s}$

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

$\zeta(s) = \prod_{p} (1 - p^{-s})^{-1}.$

Parent article