pi^2/6 and 6/pi^2

[Jamie M]

Hi,


I think there is a relationship between pi^2/6 and 6/pi^2
ie from the below:

"The Basel problem asks for the precise summation of the reciprocals of
the squares of the natural numbers" which sums to 6/pi^2

https://en.wikipedia.org/wiki/Basel_problem



And the probability that two integers m and n picked at random are
relatively prime is pi^2/6

http://mathworld.wolfram.com/RelativelyPrime.html


Here is a link to the geometric proof for 6/pi^2 as the
sum of the reciprocals of the squares of the natural numbers:

"Summing inverse squares by euclidean geometry"
http://www.math.chalmers.se/~wastlund/Cosmic.pdf

Since there is a geometric proof for 6/pi^2, by the principle
of reciprocals? hehe there should be a geometric proof for pi^2/6
being the probability that two integers m and n picked at random
are relatively prime too right?

Maybe all the geometry can just be inverted or something :D

cheers,
Jamie

[Jamie M]

Just noticed also that:

"the asymptotic density of squarefree numbers is 6/pi^2 too"

http://mathworld.wolfram.com/Squarefree.html

cheers,
Jamie

[Earle Jones27]

On 2016-03-31 12:39:21 +0000, Jamie M said:

> Hi,
>
>
> I think there is a relationship between pi^2/6 and 6/pi^2

*
The relationship is called the "reciprocal".

earle
*

[Jamie M]

How does the reciprocal relationship apply to this:

1. the probability that two integers m and n picked at random are
relatively prime is pi^2/6

2. the sum of the reciprocals of the squares of the natural
numbers is 6/pi^2 is

cheers,
Jamie

[Jamie M]

Somehow I got those two reversed, should be:

1. the probability that two integers m and n picked at random are

relatively prime is 6/pi^2

2. the sum of the reciprocals of the squares of the natural

numbers is pi^2/6

[abu.ku...@gmail.com]

thought it was the two-sixths power of pi, and
teh secondpower of six piths