[seqfan] Re: Sums of harmonic errors

[Tomasz Ordowski]

AT THE END.

Contrary to my conjecture:
https://www.wolframalpha.com/input?i=Limit%5BExp%5B-EulerGamma%5D*ExpIntegralEi%5BLog%5Bn%5D+%2B+EulerGamma%5D+-+Sum%5B1%2FHarmonicNumber%5Bk%5D%2C+%7Bk%2C+1%2C+n%7D%5D%2C+%7Bn+-%3E+Infinity%7D%5D
But is this result +oo reliable?

Similar limit, but looking more familiar:
lim_{n->oo} (li(n) - Sum_{k=1..n} 1/(H_k - gamma)) = ?
https://www.wolframalpha.com/input?i=Limit%5BLogIntegralLi%5Bn%5D+-+Sum%5B1%2F%5BHarmonicNumber%5Bk%5D+-+EulerGamma%5D%2C+%7Bk%2C+1%2C+n%7D%5D%2C+%7Bn+-%3E+Infinity%7D%5D
This page returns +oo again.
If so, for what n does this difference change sign from negative to
positive?
Notice that for small n this is negative.

Best,

Tom ORDO

czw., 13 kwi 2023 o 05:34 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> 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!
>>
>>