[seqfan] Euler product and prime zeta function

[Tomasz Ordowski]

Dear readers!

By the Euler product,
Product_{prime p} (1-1/(2p^n-1))/(1+1/(2p^n-1)) =
= Product_{prime p} (1-1/p^n) = 1/zeta(n), for n > 1.

Note that w(n) = (1-1/zeta(n)/(1+1/zeta(n) = (zeta(n)-1)/(zeta(n)+1)
is the relativistic sum of the velocities v = 1/(2p^n-1) over all primes p,
in units where the speed of light c = 1. Cf. A348829 / A348830.

So, according to the above physical interpretation,
1/(2*2^n-1) < w(n) = (zeta(n)-1)/(zeta(n)+1) < Sum_{prime p} 1/(2p^n-1).
The prime zeta P(n) = Sum_{prime p} 1/p^n ~ 2(zeta(n)-1)/(zeta(n)+1),
(but P(n) < 2w(n) for n > 1). Hence P(n) ~ log(zeta(n)) ~ zeta(n)-1.

Let a(n) be the smallest prime q such that
Sum_{prime p <= q} 1/(2p^n-1) > (zeta(n)-1)/(zeta(n)+1).

Is a(n) = Q for every n >= N (until the end of the calculation)?
If so, what are the hypothetical values of Q and N ?

Best regards,

Thomas Ordowski
__________________
Cf. A348829 / A348830.
See my draft with a new conjecture in the formula section:
The On-Line Encyclopedia of Integer Sequences® (OEIS®)
<https://oeis.org/history/view?seq=A348829&v=94>