mersenneforum.org  

Go Back   mersenneforum.org > Prime Search Projects > Prime Gap Searches

Reply
 
Thread Tools
Old 2020-08-26, 18:31   #78
kruoli
 
kruoli's Avatar
 
"Oliver"
Sep 2017
Porta Westfalica, DE

3·101 Posts
Default

Quote:
Originally Posted by mart_r View Post
The TEX expression doesn't work and I can't figure out why. Anyway it's in the graph in the attachment.
It works with the "fancy" TeX engine after messing a bit with the command:
\[(\log\frac{\#\text{gaps with }M>k}{2^{21}}+k) \cdot 2n\]
kruoli is online now   Reply With Quote
Old 2020-09-11, 02:15   #79
danaj
 
"Dana Jacobsen"
Feb 2011
Bangkok, TH

5·181 Posts
Default

Arxiv: "Primes in short intervals: Heuristics and calculations" by Granville and Lumley, 10 Sep 2020. Interesting.
danaj is offline   Reply With Quote
Old 2020-09-14, 12:57   #80
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

26·32 Posts
Default

Thanks, Dana. That's quite an interesting read indeed.

Quote:
Originally Posted by mart_r View Post
The density of gaps with merit >=M between consecutive primes appears to be on average e^{-M+\frac{a_M}{\log p}} for integer M with the following values aM
The behaviour of aM is only a prelude to a known line of probabilistic reasoning, isn't it?
(1-\frac{1}{\log x})^{M(\log x)}\hspace{3}=\hspace{3}(1-\frac{M}{2\hspace{1}\log x}+O(\frac{1}{(\log x)^2}))\hspace{1}e^{-M}<br />
(The error term may or may not be correctly applied here, but who cares...)
Huh?? Now the [$$] tags don't work properly, have to use [TEX] again.


I get the feeling this is what especially chapters 6 and 7 in 2009.05000 are pointing toward - I'm still grappling with the connections between u+-, c+-, delta+-, and sigma+- there - but let me paraphrase it in a way that I've worked out by myself. (Great, that prompted my brain to play Depeche Mode on repeat: "Let me show you the world in my eyes..." )

Using Cramér's uniformly distributed probability model, looking for a gap of size (log x)², we want to know the probability P for
(1-\frac{1}{\log x})^{(\log x)^2}, which has a series expansion e^{-(\log x+\frac{1}{2}+\frac{1}{3\hspace{1}\log x}+...)}\hspace{3}=\hspace{3}e^{-\sum_{n=1}^\infty \frac{(\log x)^{2-n}}{n}}.


Considering only odd numbers to be potential prime number candidates, this would turn into
(1-\frac{2}{\log x})^{\frac{1}{2}(\log x)^2}\hspace{3}=\hspace{3}e^{-\sum_{n=1}^\infty \frac{2^{n-1}\hspace{1}(\log x)^{2-n}}{n}}<br />


and sieving with small primes <=z, where w=\prod_{primes\hspace{1}q}^z \frac{q}{q-1},
(1-\frac{w}{\log x})^{\frac{1}{w}(\log x)^2}\hspace{3}=\hspace{3}e^{-\sum_{n=1}^\infty \frac{w^{n-1}\hspace{1}(\log x)^{2-n}}{n}}<br />


and since w ~ (log x)\hspace{1}e^{\gamma}, P would go down toward zero by "allowing" to sieve primes up to z=x^{e^{-\gamma}}
which is just about one Buchstab function away from Granville's conjecture.



What I'm not quite sure about is the way that P accumulates over the entirety of x on the number line. If I got this right, the reasoning outlined above assigns the probability to every integer respectively. But aren't we looking at intervals of size (log x)², in each of which Cramér's probability, which is asymptotic to \frac{1}{x\hspace{1}\sqrt{e}}, is in effect? The simple analogy to the series \sum_{n=2}^\infty \frac{1}{n\hspace{1}(\log n)^m} probably comes to mind, which is convergent for m>1. P is even smaller for the modified sieved versions, which in turn would mean we may never see a gap of size (log x)² between primes of the size of x.

For now, that's all there is...

Last fiddled with by mart_r on 2020-09-14 at 13:04
mart_r is offline   Reply With Quote
Old 2020-09-18, 13:40   #81
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

26×32 Posts
Default

I suppose my rambling theories are, as the saying goes, "not even wrong". Right?
mart_r is offline   Reply With Quote
Old 2020-09-18, 17:48   #82
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

5,923 Posts
Default

Quote:
Originally Posted by mart_r View Post
I suppose my rambling theories are, as the saying goes, "not even wrong". Right?
I wouldn't say that, but I don't really understand what you're trying for here. The point of the paper was to give a heuristic which significantly improves upon Cramer; why would you analyze it with Cramer's model? It's like analyzing a black hole with Newtonian physics.

Am I missing something?
CRGreathouse is offline   Reply With Quote
Old 2020-09-18, 18:22   #83
mart_r
 
mart_r's Avatar
 
Dec 2008
you know...around...

26×32 Posts
Default

I guess I'm trying to argue that there may be only finitely many gaps of length > (log x)².

So I thought there may be an error in my outlined reasoning (worth elaborating...?) that one of the brilliant minds in this forum could point out to me, or at least tell me whether I'm somewhat on the right path to further enlightenment.


Even a simple "wrong" or "right" would be better than nothing at all...

Last fiddled with by mart_r on 2020-09-18 at 18:33
mart_r is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Basic Number Theory 4: a first look at prime numbers Nick Number Theory Discussion Group 6 2016-10-14 19:38
Before you post your new theory about prime, remember firejuggler Math 0 2016-07-11 23:09
Mersene Prime and Number Theory Ricie Miscellaneous Math 24 2009-08-14 15:31
online tutoring in prime number theory jasong Math 3 2005-05-15 04:01
Prime Theory clowns789 Miscellaneous Math 5 2004-01-08 17:09

All times are UTC. The time now is 09:34.

Thu Sep 24 09:34:12 UTC 2020 up 14 days, 6:45, 0 users, load averages: 1.25, 1.40, 1.38

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.