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 - 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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File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID

993: sh

Please specify a *single* volume group to restore.