Jérémy Champagne, University of Waterloo
Equidistribution and the probability of coprimality of some integer tuples
" What is the probability of two random integers being coprime? "
This question, sometimes called " Chebyshev’s Problem », is very natural and happens to have a very straightforward answer. Using only elementary methods, one can easily show that the natural density of pairs (m,n) with gcd(m,n)=1 is exactly 1/zeta(2)=6/pi^2=60.8..%.
Knowing this, one might seek certain g:N->N for which the density of n’s with gcd(n, g(n))=1 is also 1/zeta(2), which give a certain sense of randomness to the function g. Many functions with that property can be found in the literature, and we have a special interest for those of the form g(n)=[f(n)] where f is a real valued function with some equidistributive properties modulo one; for example, Watson showed in 1953 that g(n)=[αn] has this property whenever α is irrational. In this talk, we use a method of Spilker to obtain a more general framework on what properties f(n) must have, and also what conditions can replace coprimality of integer pairs.
MC 5403