# User:Guardian of Light

User at Wikipedia.

## Major Contributions

Some pages I've invested a lot in:

## Creations

The pages I myself have made (from ye olde scratch) are--in chronological order:

## Problems

To show the probability that two integers chosen at random are relatively prime is ${\displaystyle {6 \over \pi ^{2}}}$.

Proof: It is sufficient to show ${\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}={\pi ^{2} \over 6}}$. When we have a polynomial with constant term one, we may rewrite it in factored form as follows: If ${\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{r}\,}$ are the roots of a polynomial p(z), then we may write ${\displaystyle p(z)=\left(1-{z \over \alpha _{1}}\right)...\left(1-{z \over \alpha _{r}}\right)}$.

Now examine the power series for the function sin(z)/z. ${\displaystyle {\sin z \over z}=1-{z^{2} \over 3!}+{z^{4} \over 5!}+...+{(-1)^{n}z^{2n} \over (2n+1)!}}$

Well we also know we can rewrite sin(z)/z in terms of its roots to be:

${\displaystyle \left(1-{z \over \pi }\right)\left(1+{z \over \pi }\right)\left(1-{z \over 2\pi }\right)\left(1+{z \over 2\pi }\right)\left(1-{z \over 3\pi }\right)\left(1+{z \over 3\pi }\right)...\left(1-{z \over k\pi }\right)\left(1+{z \over k\pi }\right)...}$

If we examine the quadratic term in each we find that:

${\displaystyle {1 \over 3!}={1 \over \pi ^{2}}\sum _{n=1}^{\infty }{1 \over n^{2}}\rightarrow {\pi ^{2} \over 6}=\sum _{n=1}^{\infty }{1 \over n^{2}}{\text{Q.E.D.}}}$