To show the probability that two integers chosen at random are relatively prime is .
Proof: It is sufficient to show .
When we have a polynomial with constant term one, we may rewrite it in factored form as follows:
If are the roots of a polynomial p(z), then we may write .
Now examine the power series for the function sin(z)/z.
Well we also know we can rewrite sin(z)/z in terms of its roots to be:
If we examine the quadratic term in each we find that:
To show the probability that two integers chosen at random are relatively prime is .
Proof: It is sufficient to show .
When we have a polynomial with constant term one, we may rewrite it in factored form as follows:
If are the roots of a polynomial p(z), then we may write .
Now examine the power series for the function sin(z)/z.
Well we also know we can rewrite sin(z)/z in terms of its roots to be:
If we examine the quadratic term in each we find that: