[seqfan] Re: New twin prime constant t and the relativistic sum w of 2/(p+q)

[Ami Eldar]

Hello,

The sum of 2/(p+q) is also the sum of 1/((p+q)/2), i.e., the sum of the
reciprocals of the averages of the twin prime pairs,
0.92883582713... (A241560).

Best,
Amiram


On Sat, Jan 29, 2022 at 6:59 PM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> Dear readers!
>
> The sum of 2/(p+q) over all pairs of twin primes p,q is convergent.
> What is the approximate value of this sum?
> Is this constant in the OEIS?
>
> A more difficult issue. Let's define:
> Relativistic sum w of the velocities 2/(p+q) over all twin prime pairs
> p,q.*
> Let t = (1-w)/(1+w) = (3/5)(5/7)(11/13)... = lim_{n->oo} t(n) = ?
> where t(n) = Product_{k=1..n} (A001359(k) / A006512(k)).
> Hence w = (1-t)/(1+t) = ??
>
> I expected that 1/t < e^2. This is provable.
> Amiram Eldar noticed that t = exp(-A331370).
> We have 1/t = e^1.8721788... =  6.5024485...
> See the constant A331370 - OEIS <https://oeis.org/A331370>
> The constants t = 0.15378822...
> and w = 0.73342036...
>
> Best regards,
>
> Thomas Ordowski
>  _______________
> (*) In physical units where the speed of light c = 1.
> Cf. A348829 - OEIS <https://oeis.org/A348829> / A348830 - OEIS
> <https://oeis.org/A348830>
>
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