On 12/3/06, Daniel Asimov <dasimov@earthlink.net> wrote:
... ------------------------------------------------------------------ Let f(N) be the probability that 4 random integers i,j,k,m in the range 1 <= i,j,k,m <= N satisfy
gcd(i,j) = gcd(k,m) .
Find the limit of f(N) as N -> oo. ------------------------------------------------------------------
Well, if Dan does have a \pi-free argument, I'll be interested to see it. But I now have a \mu-free one, so here it is [a long way down, to be sure]. Let S(p, q) denote the subset of the lattice S of integer vectors [i,j,k,l], such that hcf(i,j) = p and hcf (k,l) = q; and let c denote the density of S(1, 1). Then S is the direct union of the S(p,q), so 1 = \sum_{p,q} c / p^2 q^2 = c \zeta(2)^2 = c \pi^2 / 36; the problem demands the density of the direct union of the S(p,p), = \sum_{p} c / p^4 = c \zeta(4) = c \pi^2 / 90, = (c \pi^2 / 36) 2/5 = 2/5 by the above. Fred Lunnon