[seqfan] The constants B_3 and b_3, as well as B = 1

[Tomasz Ordowski]

Dear readers!

As is well known, the constant (related to Mertens' first theorem):
B_3 = lim_{x->oo} (log(x) - Sum_{p<=x} log(p)/p) = 1.332582...,
where p is a prime.
Cf. https://oeis.org/A083343 and the last formula in
https://mathworld.wolfram.com/MertensConstant.html
Let's define an analogous constant, namely (without proof):
b_3 = lim_{x->oo} (log(x) - Sum_{p<=x} (q - p)/p) = 0.0477967..,
where q is the next prime after p, so (q - p) is a prime gap.
The value of b_3 was calculated by Amiram Eldar.
Since (q - p)/p = q/p - 1, we have
b_3 = lim_{x->oo} (pi(x) + log(x) - Sum_{p<=x} q/p),
where q/p is a prime ratio.
Hence Sum_{p<=x} q/p = pi(x) + log(x) - b_3 + o(1).
Is this asymptotic known or provable?

Finally, a somewhat similar but easier question:
does lim_{x->oo} (log(x) - 1/pi(x) Sum_{p<=x} log(p)) = 1 ?
In other words, lim_{n->oo} (log p(n) - 1/n Sum_{k=1..n} log p(k)) = 1,
and equivalently, lim_{n->oo} (p(n)/n - 1/n Sum_{k=1..n} log p(k)) = 0 ?
In forms more convenient for numerical calculations.

Maybe someone will find the answers.
But first, please see the footnotes.

Best regards,

Thomas Ordowski
_________________
Mertens' first theorem (in modern notation):
Sum_{p<=x} log(p)/p = log(x) + O(1).
Cf. https://en.wikipedia.org/wiki/Mertens%27_theorems
and https://pl.wikipedia.org/wiki/Twierdzenia_Mertensa
"Ordowski's last conjecture" (in this notation):
Sum_{p<=x} (q/p - 1) = log(x) + O(1),
where q is the next prime after prime p.
_________________________
The answer to the last question seems to be YES.
This is a result of the proven fact that the present
value of the historical Lagendre's constant B = 1.
If pi(x) = x / (log(x) - B(x)),
then B = lim_{x->oo} B(x) = lim_{x->oo} (log(x) - x/pi(x)) = 1.
See: https://en.wikipedia.org/wiki/Legendre%27s_constant