20200826, 18:31  #78 
"Oliver"
Sep 2017
Porta Westfalica, DE
3·101 Posts 

20200911, 02:15  #79 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
5·181 Posts 
Arxiv: "Primes in short intervals: Heuristics and calculations" by Granville and Lumley, 10 Sep 2020. Interesting.

20200914, 12:57  #80  
Dec 2008
you know...around...
2^{6}·3^{2} Posts 
Thanks, Dana. That's quite an interesting read indeed.
Quote:
(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 , which has a series expansion . Considering only odd numbers to be potential prime number candidates, this would turn into and sieving with small primes <=z, where , and since w ~ , P would go down toward zero by "allowing" to sieve primes up to 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 , is in effect? The simple analogy to the series 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 20200914 at 13:04 

20200918, 13:40  #81 
Dec 2008
you know...around...
2^{6}×3^{2} Posts 
I suppose my rambling theories are, as the saying goes, "not even wrong". Right?

20200918, 17:48  #82  
Aug 2006
5,923 Posts 
Quote:
Am I missing something? 

20200918, 18:22  #83 
Dec 2008
you know...around...
2^{6}×3^{2} Posts 
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 20200918 at 18:33 
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