Puzzle 1: Solution

Q     Natural numbers a,b,c,d between 1 and a xillion are chosen at random, with each of the xillion-to-the-fourth-power possible quadruples equally likely. What is the probability that gcd(a,b)=gcd(c,d) ?

A     Exactly 40% !

This number arises as zeta(4)/zeta(2)2. For each n=1,2,3,..., the probability that gcd(a,b)=n is n-2/zeta(2), so the probability that gcd(a,b)=gcd(c,d)=n is n-4/zeta(2)2; now sum over n.

Can the answer 2/5 be proved by elementary means?

This can be regarded as a form of a question attributed to Wagstaff in problem B48 of R.K.Guy's Unsolved Problems in Number Theory (Springer, 1981): “Wagstaff asked for an elementary proof (e.g., without using properties of the Riemann zeta-function) that [the product of (p2+1)/(p2-1) over all primes equals 5/2].”
Is there any value of a xillion for which the probability is actually less than 40%?