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

Have there been numerical experiments comparing
pi(B) - pi(A) to a Poisson distribution,
or something similar?

David Bernier

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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