a repository of mathematical know-how

Order by property

Stub iconThis article is a stub.This means that it cannot be considered to contain or lead to any mathematically interesting information.

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


Post new comment

(Note: commenting is not possible on this snapshot.)

Before posting from this form, please consider whether it would be more appropriate to make an inline comment using the Turn commenting on link near the bottom of the window. (Simply click the link, move the cursor over the article, and click on the piece of text on which you want to comment.)