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Topic: number of primes in short interval [a, b] similar to Poisson?
Replies: 1   Last Post: Feb 6, 2013 7:33 AM

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David Bernier

Posts: 3,892
Registered: 12/13/04
Re: number of primes in short interval [a, b] similar to Poisson?
Posted: Feb 6, 2013 7:33 AM
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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 - 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?

dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.

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