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 benefitial 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 set $A_h (h = 0,1,2...)$ 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

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

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