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Order by property

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Quick description

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


Some real analysis.

Example 1

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}.

General discussion