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Re: number of primes in short interval [a, b] similar to Poisson?
Posted:
Feb 6, 2013 7:33 AM


On 02/06/2013 07:12 AM, David Bernier wrote: > 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.
It would be simpler to have f(t) = K, a constant. For n>= 10^9, say K=100.
So A(n) = n and B(n) = n + floor(100 log(n)).
Let s_n = pi(B(n))  pi(A(n)).
Letting n vary from 10^9 to 10^9 + 10^6  1 gives us 10^6 values of s_n.
Do the 10^6 values of s_n "conform" to a Poisson r.v. of parameter 100?
You may place your bets.
> Have there been numerical experiments comparing > pi(B)  pi(A) to a Poisson distribution, > or something similar?
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