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The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems:

$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{n=1}^{n=k} \frac{x_{n}}{n} \\ \text{subject to constraints:} & k + \displaystyle\sum_{n=2}^{n=k}x_{n}=1 \\ & x_1 \geq -1 \end{array}$$

for all $k$ and for $n>1:$

$$-2(n-1) \leq x_n \leq 0 \tag{4}$$

That is, there is one linear programming problem for each $k$.

Based on a OEIS search, the solutions $f(k)$ to the linear programming problems (without the first column) appear to have the asymptotic:

$$f(k)=-\left(2 \sqrt{k}-2 \log \left(\sqrt{k}+1\right)-2 \gamma +1\right) \tag{5}$$ Is it true?

Please don't be so harsh on me. If the problem is ill defined in the latex I post the short Mathematica program from which I defined the optimization problem.

(*start*)
nn = 180;
TableForm[
  L2 = Table[
    LinearProgramming[
     Table[1/n, {n, 1, k}], {Table[If[n == 1, k, 1], {n, 1, k}]}, {{1,
        0}}, Table[
      If[n == 1, {-1, 1}, {-2 (n - 1), 0 (n - 1)}], {n, 1, k}]], {k, 
     1, nn}]];
t1 = Table[Sum[L2[[n, k]]/k, {k, 2, n}], {n, 2, nn}];
t2 = Table[-(2*k^(1/2) + 1 - 2*Log[k^(1/2) + 1] - 2*EulerGamma), {k, 
    2, nn}];
Show[ListLinePlot[t1], ListLinePlot[t2, PlotStyle -> Red]]
ListLinePlot[t1/t2]

The blue curve is the linear programming minimum and the red curve is the asymptotic. asymptotic and linear programming minimum

Zoom in:
enter image description here

The ratio between the linear programming minimum and the asymptotic tends to one. ratio of blue and red curve

So as I said this is NOT a bound on the partial sums of the Möbius inverse of the Harmonic numbers.

The solutions $x_n..x_k$ to the $k$-th linear programming problem form a number triangle:

$$\begin{array}{llllllllllll} 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -1 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -2 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -3 & 0 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -4 & 0 & 0 & 0 & 0 & \text{} & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -4 & -1 & 0 & 0 & 0 & 0 & \text{} & \text{} & \text{} & \text{} \\ 1 & -2 & -4 & -2 & 0 & 0 & 0 & 0 & 0 & \text{} & \text{} & \text{} \\ 1 & -2 & -4 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & \text{} & \text{} \\ 1 & -2 & -4 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \text{} \\ 1 & -2 & -4 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}$$

The first column is here equal to the all ones sequence because Mathematicas linear programming command seems to require it. But setting the constraint to begin with $k$ (in the linear program in the beginning of the question) makes it equivalent to the first column in the numerators for partial sums of the Möbius inverse of the Harmonic numbers.

Counting only the negative entries in each row we find with a OEIS search that their number appear to be nearest integer to square root of $n$, and from there it becomes easy to conjecture formula $(5)$.

Notice also that for all $k$:

$$\left\lfloor k^{\text{epsilon}+\frac{1}{2}}+\frac{1}{2}\right\rfloor=\left\lfloor \sqrt{k}+\frac{1}{2}\right\rfloor$$ holds for sufficiently small epsilons.

The partial sums of the Möbius inverse of the Harmonic numbers have the numerators:

$$J(m,k)=\begin{array}{lllllll} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & -2 & 0 & 0 & 0 & 0 \\ 4 & -1 & -1 & -1 & 0 & 0 & 0 \\ 5 & 0 & 0 & 0 & -4 & 0 & 0 \\ 6 & -1 & -2 & -1 & -3 & 2 & 0 \\ 7 & 0 & -1 & 0 & -2 & 3 & -6 \end{array}$$

given by the sum:

$$\sum _{n=1}^m \text{If}[n\geq k,a(\gcd (n,k)),0]$$

for
$n=1..m$,
$k=1..N$,
$m=1..N$. and where $a(n)$ is the Dirichlet inverse of the Euler totient function.

The properties are:

$$\sum_{k=1}^{k=n} \frac{J(n,k)}{k}=\lim_{s\to 1+} \sum_{m=1}^{m=N}H_m \sum_{d\mid m}\frac{\mu(d)}{d^{s-1}}$$ which is the partial sums of the Möbius inverse of the m-th harmonic number

$$\sum_{k=1}^{k=n}J(n,k)=1$$ as in the constraint in the linear programming problem. $$J(n,1)=n$$ as in the linear programming problem (but in the linear programming problem it is in the constraint and not the goal function because of some Mathematica technicality.)

The last property, for all $n$:

$$-2(k-1) \leq J(n,k) \leq 2(k-1)$$

is conjectural and differs from the linear programming problem. This last conjectural property should not be too hard to prove.

(*Numerators of the partial sums of the Möbius inverse of the \
Harmonic numbers*)(*start*)
Clear[T, n, k, a];
nn = 7;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]
TableForm[
 M = Table[
   Table[Sum[If[n >= k, a[GCD[n, k]], 0], {n, 1, m}], {k, 1, nn}], {m,
     1, nn}]]
(*end*)

Previously asked yesterday at Mathematics stack exchange, where I was not understood. I also asked about the notation at Mathematica stack exchange.

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