Thanks, Dana. That's quite an interesting read indeed.
Quote:
Originally Posted by mart_r
The density of gaps with merit >=M between consecutive primes appears to be on average for integer M with the following values a _{M}

The behaviour of a
_{M} is only a prelude to a known line of probabilistic reasoning, isn't it?
(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...