[seqfan] Re: The constants B_3 and b_3, as well as B = 1
Tomasz Ordowski
tomaszordowski at gmail.com
Wed Mar 8 07:37:09 CET 2023
P.S. However, my last question is hard,
which I explain (in a new footnote),
and I ask again:
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 ?
Maybe someone knows.
Best regards,
Thomas Ordowski
_________________________
The last two limits differ by one.
This is a result of the proven fact that the present
value of the historical Legendre'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
So we have B = lim_{n->oo} (log p(n) - p(n)/n) = 1.
pt., 3 mar 2023 o 20:05 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):
> 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
>
>
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