[seqfan] Harmonic Constant

[Tomasz Ordowski]

Dear readers!

As is well known, log(x) - x/Li(x) ~ 1.
It has also been proven that  log(x) - x/pi(x) ~ 1.
Cf. https://en.wikipedia.org/wiki/Legendre%27s_constant
Note that Sum_{k=1..n} 1/H(k) ~ Sum_{k=2..n} 1/log(k) ~
~ Integral_{2..n} dx/log(x) = li(n) - li(2) = Li(n) ~ pi(n).
So we have  A096987(n) / A124432(n) ~ pi(n).
See my draft (new comment and my formula):
https://oeis.org/history/view?seq=A096987&v=29

Let's finally define this Harmonic Constant:
H = lim_{n->oo} (log(n) - n / Sum_{k=1..n} 1/H(k)),
where the harmonic number H(k) = 1+1/2+...+1/k.

Conjecture:
the value of this constant H = 1 - gamma,
where gamma = 0.577... is Euler's constant.

Equivalently:
lim_{n->oo} (H(n) - n / Sum_{k=1..n} 1/H(k)) = 1.
Nice!

Is this a known result or provable?
Numerical verification will be hard.*

Best regards,

Thomas Ordowski
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(*) Calculations up to n = 10^15 should give the desired result,
as suggested by a numeric sample from Amiram Eldar. Thanks!