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Daniel Asimov
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What is the expected size of the complement of the union of random cosets of the prime ideals of ℤ?

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 - (C2C3C5 ∪ ...)).

What is the expected value of S ?

(In the simplest case with all Xp = 0, we get S = 2.)

Daniel Asimov
  • 2.9k
  • 24
  • 26