For each rational prime p let Xp denote the random variable uniformly distributed in {0, 1, ..., p-1} with all the Xp independent of each other. Define the coset Cp of the prime ideal pℤ via
Cp = pℤ + Xp.
Let S be the random variable equal to the size of the complement of the union of all the Cp:
S = card(ℤ - (C2 ∪ C3 ∪ C5 ∪ ...)).
What is the expected value of S ?
(In the simplest case with all Xp = 0, we get S = 2.)