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number of primes in short interval [a, b] similar to Poisson?
Posted:
Feb 6, 2013 7:12 AM


If f: R> R is a nice, increasing function of t with f(10)>0 (say) and f(t) <= sqrt(t) for t>=10, with f unbounded on [10, oo),
and n is some positive integer, a heuristic is that 1/log(n) integers near n are primes.
For definiteness, say f(t):= log(t).
Let A: n B:= n + floor( f(n)log(n) ) . Then BA is about f(n)log(n) and under the heuristic, would contain about f(n) primes. Here, f(t):=log(t).
So pi(B)  pi(A) ~= log(n) as an approximation, where pi is the prime counting function.
Another heuristic is that the arrival time of primes (time = prime number value) resembles a Poisson process.
Then, pi(B)  pi(A) would resemble a Poisson random variable, of parameter log(n). Mean = Poisson parameter = log(n).
I'm interested in large deviations, say 3 sigma or more from the mean log(n).
In practive, I suppose one could give the values 10^9, ... 10^9 + 10^6  1 to n and averageout.
Have there been numerical experiments comparing pi(B)  pi(A) to a Poisson distribution, or something similar?
David Bernier
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