But the primes are obviously not random, independently of the Riemann hypothesis - they're determinate consequences of the number system. Maybe I missed your point
P(X = p > 0),
that is, the probability that X = p is positive.
So, with X, the prime numbers are random.
References:
With TeX markup, polished details on measure theory are in
H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \
Walter Rudin, {\it Real and Complex Analysis,\/} ISBN 07-054232-5, McGraw-Hill, New York, 1966.\ \
and polished details on probability theory based on measure theory are in
Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia, 1992.\ \
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco, 1965.\ \
Sam, just a minute. Let me draw a sample. Got one: X = 7.
But, Joe 7 is not very surprising or interesting. Is there anything else?
Sam, sure, one more minute. Got one!
Joe, well, then, WHAT is it????
Sam, sorry, it has 2^12345 digits and will take a while to print it out; this morning I have a coffee shop meeting with Susan; and I don't want to miss Susan, cute, pretty, sweet, smart, darling, adorable, precious ..., and single!
Exercise: Show that there is a random variable Y with the same distribution as X and such that X and Y are independent random variables. I.e., knowledge of X tells us nothing about Y.
The intuitive concept of random is closer to unpredictable given even everything else, that is, what probability theory defines as independent.
There are more details in Royden, Rudin, Breiman, and Neveu. To preview, there is a non-empty set Omega with a collection F of subsets that form a sigma algebra and a measure P on F. Then random variable X is a measurable function from Omega to the set of all prime numbers. So, for some point w in Omega and function X, X(w) is a prime number. Can think of w as a trial.
Uh, this morning I'm working on some Rexx code, so here I can't reproduce or compete with the references by Royden, Rudin, Breiman, and Neveu.
Replace
P(X = p > 0)
with
P(X = p) > 0
I’m curious what this does, cause it seemingly changes nothing (tested on mobile, with “desktop version” too).