Relatively Prime Numbers and Zeta(2)

radius:7

expansion:1

p

Two integers are relatively prime if they share no common positive factors (divisors) except 1. In the graphic the points have integer

x

and

y

coordinates. If an integer pair consists of relatively prime numbers, no other such point lies on the line between the origin and this point.

With no expansion, the distribution of relatively prime pairs is shown and their density (or probability) is expressed as

p

. When this distribution is expanded by

s

, their density is reduced by the square of

s

. Because there is no intersection between the expansion sets, the individual densities can be summed to obtain the overall density, which is 1. Therefore

p

is obtained as

-1

(1+1/4+1/9+⋯)

, which is equal to

6/

2

π

.