Possible relation between Catalan’s, Ramanujan-Soldner’s and Brun’s constants
 

Hello guys,
I’d like to encourage you to help to look for the possible explanation of the very interesting formula I have “discovered” some time ago. Goes about the hypothetical relationship between the three constants (Catalan’s, Ramanujan-Soldner’s and Brun’s) related generally with the distribution of primes.
The relation is completely surprising and unexpected (neither me nor, few experts in number theory couldn't find the reason for its existence) but amazingly simple, and in some way very elegant :-), and this is why I believe could be real...
I wrote hypothetical because I’ve discovered it (not derived) a little by accident, exploring some other areas.
So, the relation results in receiving the closed formula for Brun's constant, which direct computation is not possible so far and is based on the extrapolation of achievable computational results made with the help of the twin primes conjecture.
Basing on these extrapolations (for the n up to 10^16) the value of the constant is about 1.9021605831..., and my result is arbitrarily more precise.

Here you can find some short extract of the way how the constant is calculated [url]http://numbers.computation.free.fr/Constants/Primes/twin.html[/url]
The more details you can find in Marek Wolf's works.

And here’s the formula:
Let G be the Catalan’s constant, equal to β(2) (Dirichlet beta function for s=2)
G = β(2) = 0.91596559417721901505460351493238411077414937…
Let µ be the Ramanujan-Soldner constant
µ = 1.45136923488338105028396848589202744949303228…
Let B2 be the Brun’s constant

Conjecture:
[B]B2= [8+40(µ-G)]/[16-(µ-G)][/B] = 1.9021605831029730799822614917574361…

So, simplifying and substituting (µ-G) with an A, we’ve got:

[B]B2=(8+40A)/(16-A)[/B]

Maybe this formula is only the coincidence, but its unusual form suggest it might be true, and this is why I’d like to share this “discovery” with you, because maybe somebody of you would be able to notice something that could help to find its explanation - maybe it could also shed the new light on the problem of distribution of the twin (and other constellations) primes.

Kind regards,
Marcin

[QUOTE=MarcinLesniak;495065]I'm very curious about your opinion about it :-)[/QUOTE]

Is it just me, or are we seeing a whole lot of this kind of thing recently?

It's a bit like a friend of mine said about guys coming into his lab claiming to have found a wonderful short-cut for the FFT, and then his people having to patiently explain why the short-cuts wouldn't actually work for almost all cases, and those where it did work were already known....

[QUOTE=chalsall;495068]Is it just me, or are we seeing a whole lot of this kind of thing recently?
[/QUOTE]
It is in a PDF, so it [I]must[/I] be true!

[QUOTE=MarcinLesniak;495065]

[B]B2=(8+40A)/(16-A)[/B] [/QUOTE]

aka [TEX]B2=\frac{1+5A}{2-\frac{A}{8}}=(-40)+\frac{81}{2-\frac{A}{8}}[/TEX]

Guys,
maybe, just to respect some other readers' time let's limit ourselves to some more substantive comments, OK? When you have nothing valuable to write/add, just leave without unnecessary extending the subject.

If you expect that posting here will bump the information content from having it posted on NMBRTHRY list on August 05, you will be disappointed.

On NMBRTHRY, perhaps three dozen math professors and associate professors read your post. Here - ... three and a half people without any degrees and some not even knowing any maths?

Of the three, the constant known with the least precision is Brun's constant.

The value often quoted is 1.902160583104... but from [URL="http://numbers.computation.free.fr/Constants/Primes/twin.html"]Sebah's notes about his computation[/URL] it's clear that the precision is at best only about 1.902160583... The digits aren't calculated directly, rather, it's an extrapolation, see the graph in the link.

Sebah did the computations leading to this estimate way back in 2002, using twin primes up to 10[SUP]16[/SUP]. Surely today's computers can do a little better, and maybe add a data point or two to the graph.


As far back as 2008, Tomás Oliveira e Silva calculated the number of twin primes up to 10[SUP]18[/SUP]. I don't know the algorithm for finding pi2(x)... does it work by actually identifying the twin prime pairs, or it is some magic like the algorithm that lets you calculate pi(x) up to 10[SUP]27[/SUP] without counting individual primes.

Has there really been no progress since 2008?


[URL="https://oeis.org/A065421"]A065421[/URL]: Brun's constant
[URL="https://oeis.org/A007508"]A007508[/URL]: number of twin primes below 10[SUP]n[/SUP]

@Batalov
:-) I had a shade of hope that I could find somebody passionate about this (or closely related) subject for whom such (possible) relation could prompt some direction of explorations :-).

@GP2
Has there really been no progress since 2008?

Unfortunately as far as I know, not...