Could you offer a citation? I tried to find the result you mention, and I'm at a loss to even pin down a definition of "random distribution" that might be used.
The closest I can come is the coincidence that a random variable is used in the proof of the Prime Number Theorem, but that's just a modeling technique.
At least I did find this awesome quotation, which seems to play both sides:
In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).
The closest I can come is the coincidence that a random variable is used in the proof of the Prime Number Theorem, but that's just a modeling technique.
At least I did find this awesome quotation, which seems to play both sides:
In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).
http://mathworld.wolfram.com/PrimeNumber.html