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Probability of being coprime (Posted on 2018-07-11) Difficulty: 3 of 5
What is the probability of two randomly chosen integers being coprime?

Bonus(d4): How does the result relate fo the density
of square-free numbers?

No Solution Yet Submitted by Ady TZIDON    
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Coprimality and the Basel problem Comment 4 of 4 |
A little research shows that, yes, the coprimality of 2 random integers and the infinite sum of the squares of the positive integers are related. The following is extracted from

Informally, the probability that any number is divisible by a prime (or in fact any integer) p is 1/p. For example, every 7th integer is divisible by 7. Hence the probability that two numbers are both divisible by p is 1/p^2. One is led to guess that the probability that two numbers are coprime is given by a product over all primes:

Prod {over prime p} ( 1- 1/p^2) = Prod {p} (1 / (1 - p^-2)^-1

                                                = 1/ζ(2)

Here ζ refers to the Riemann Zeta Function which Euler showed in 1735 sums to π2/6.

Who woulda thunk it?

Edited on July 16, 2018, 1:25 pm

Edited on July 17, 2018, 8:10 pm

Edited on July 17, 2018, 8:11 pm
  Posted by JayDeeKay on 2018-07-16 13:19:15

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