Re: recent paper on prime gaps: p_{n+1} - p_n

Chip Eastham wrote:
On May 25, 1:47 pm, David Bernier<david...@xxxxxxxxxxxx> wrote:
[...]

I did that with the PARI command:

@ (04:35) gp> for(X=1,10000000,
if(ispseudoprime(a+X+X+1),write("G:\\psprimes10meg", X+X+1)))

// pseudoprimes from 10^100 to 10^100 + 20,000,000.

It found 86,794 pseudoprimes in about 55 minutes. The average gap near
10^100 is expected to be log(10^100), where all prime gaps between
10^100 and (say) 10^100 + 10^20 count once...

So 1 hour: 20 million tested for being pseudoprime near 10^100.

There's another histogram from the gaps/(average gap); There are gaps of
6*average and even more, but they don't leave much of a trace on the
histogram figure...

http://www.geocities.com/ezcos/gapsgoogol76k.jpg

The probabilistic model that has been proposed is the (continuous)
exponential distribution
with rate parameter lambda = 1.

Cf.: Probability density function graphs, etc. at:http://en.wikipedia.org/wiki/Exponential_distribution

Note: (log(10^100))^2 ~= 53,000 ; this gives an indication of what
"large" gaps might mean in
the Soundararajan paper, and the literature on large gaps ,
or rather, conjectures on large gaps.

David Bernier

Hi, David:

I was puzzled by the legend on your graphs of
"normalized" prime gaps, but from (gasp! reading
the PDF you linked at the beginning of thread,
I take it to mean gaps (near x) divided by log x.

Hence the need to tabulate by a histogram, right?

For primes around 10^100, the average gap is 100*log(10) or 230.3 .

There is one gap of length 2280 for the 2nd set of data:
(15:03) gp > isprime(10^100 + 3188991)
%51 = 1

(22:28) gp > nextprime(10^100 + 3188992)-10^100
%53 = 3191271

3191271- 3188991 = 2280.

The histogram was produced using Matlab, and the number of bins
was set to 100 by me. By "normalized gap", I mean
(p_{n+1}- p_n)/(100*log(10)) for the p_n near 10^100.

We could also have files x.dat with 2, 4, ... 2280 and
y.dat with the counts of gaps with lengths 2, ... 2280.
Then, plot(x, y) would give a rather jagged graph; it seems to
me the top of the figure would resemble the top of a histogram with
about 1140 bins ...

The Cramer model assumes Prob(x is prime) and Prob(x+2 is prime)
are independent and an analysis of the expected correction is
in a paper by Goldston (of quite general interest) here:
http://arxiv.org/abs/0710.2123 (title on arxiv should include twin primes).
[ Thanks to Rich Burge for telling me about this paper.]

Because the Cramer model is not very accurate, I've been
wondering what the average value of:
[ (p_{n+1} - p_n)/( log(p_n)) ]^2 should be .

From reading Goldston's paper, I get the impression that
some researchers in the area are thinking about trying
to prove "bounded gaps" sooner or later.
That would be: liminf_{n->infinity} (p_{n+1} - p_n) = K < infinity .


David





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