[seqfan] Re: Sums of harmonic errors

[Tomasz Ordowski]

P.S. Conjecture:
E = exp(-gamma) = A080130 = 0.56145948356688516982414321479...
Sum_{k=1..n} 1/H_k = exp(-gamma) (Ei(log(n) + gamma) - 1) + o(1).
Cf. A096987 : https://oeis.org/history/view?seq=A096987&v=98

T. Ordowski
___________________
https://oeis.org/A080130
https://oeis.org/A096987

śr., 5 kwi 2023 o 18:42 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> Hello everyone again!
>
> Alabdulmohsin (2018) derived closed-form expression for the sum [sic]:
> Sum_{n >= 1} (log(n) + 1/(2n) + gamma - H_n) = (log(2 pi) - 1 - gamma)/2.
> Cf. https://en.wikipedia.org/wiki/Euler%27s_constant#Asymptotic_expansions
> The exact value of a similar sum is easier to determine, namely:
> Sum_{n >= 1} (H_n - log(n+1/2) - gamma) = (2 gamma + 1 - log(8))/2.
> With estimate: 1/(24(n+1)^2) < H_n - log(n+1/2) - gamma < 1/(24n^2).
> Cf. https://en.wikipedia.org/wiki/Euler%27s_constant#Series_expansions
>
> I also calculated approximate values of other sums:
> Sum_{n >= 1} (1/H_n - 1/(log(n) + 1/(2n) + gamma)) = 0.0867629...
> C = Sum_{n >= 1} (1/(log(n+1/2) + gamma) - 1/H_n) = 0.0229825...
> Problem: are there closed-form expressions for these constants?
> Cf. A096987 (see the second formula):
> https://oeis.org/A096987
> See again A096987 (the third formula).
> Note that Integral dx / (log(x) + gamma) =
> = exp(-gamma) Ei(log(x) + gamma) + c,
> where Ei(x) is the exponential integral function of real x,
> and as is well known, Ei(log x) = li(x), which looks familiar.
> Let's rewrite my third formula in a slightly more precise notation:
> Sum_{k=1..n} 1/H_k = exp(-gamma) Ei(log(n) + gamma) - E + o(1).
> Hence we have the following definition to compute of my new constant
> E = lim_{n->oo} (exp(-gamma) Ei(log(n) + gamma) - Sum_{k=1..n} 1/H_k).
> Hard task: find a good numerical approximation of this constant (if it
> exists).
>
> Best,
>
> Thomas
> _________________
> See also my theorem with a short proof
> at the end of my new entry to A096987:
> https://oeis.org/A096987 (numerators) /
> https://oeis.org/A124432 (denominators),
> where is also my new comment. Thanks!
>
>