```Date: Feb 6, 2013 7:33 AM
Author: David Bernier
Subject: Re: number of primes in short interval [a, b] similar to Poisson?

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 B-A 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 average-out.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 - 1gives 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?-- dracut:/# lvm vgcfgrestoreFile descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh   Please specify a *single* volume group to restore.
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