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

Tomasz Ordowski tomaszordowski at gmail.com
Fri Jan 28 15:15:22 CET 2022

```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
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(*) In physical units where the speed of light c = 1.
Cf. A348829 - OEIS <https://oeis.org/A348829> / A348830 - OEIS
<https://oeis.org/A348830>

```