1
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Let:

$$a(n) = \lim\limits_{s \rightarrow 1} \frac{\zeta(s)\sum\limits_{d|n} \mu(d)(e^{d})^{(s-1)}}{n}$$

$$a(n) = 1,-\frac{1}{2},-\frac{2}{3},-\frac{1}{4},-\frac{4}{5},\frac{1}{3},-\frac{6}{7},-\frac{1}{8},-\frac{2}{9},\frac{2}{5},-\frac{10}{11},\frac{1}{6},-\frac{12}{13},...$$

$$\text{a(n) = Dirichlet Inverse of the Euler Totient divided by n}$$

And: $$T(n,k) = a(GCD(n,k))$$

$$\text{GCD = Greatest Common Divisor}$$

This is a matrix starting:

$$\displaystyle T = \begin{bmatrix} +1/1&+1/1&+1/1&+1/1&+1/1&+1/1&+1/1&\cdots \\ +1/2&-1/2&+1/2&-1/2&+1/2&-1/2&+1/2 \\ +1/3&+1/3&-2/3&+1/3&+1/3&-2/3&+1/3 \\ +1/4&-1/4&+1/4&-1/4&+1/4&-1/4&+1/4 \\ +1/5&+1/5&+1/5&+1/5&-4/5&+1/5&+1/5 \\ +1/6&-1/6&-2/6&-1/6&+1/6&+2/6&+1/6 \\ +1/7&+1/7&+1/7&+1/7&+1/7&+1/7&-6/7 \\ \vdots&&&&&&&\ddots \end{bmatrix}$$

Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?

For example, the matrix size in the columns when a eigenvalue is common to the largest eigenvalue in the matrix with a size equal to row index, is:

$$\begin{array}{llllllllllllllllllll} 1 & 0 & 3 & 5 & 7 & 8 & 9 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ 0 & 2 & 14 & 26 & 27 & 28 & 29 & 30 & 31 & 32 & 34 & 35 & 36 & 37 & 38 & 39 & 40 & 41 & 42 & 43 \\ 0 & 3 & 39 & 40 & 41 & 42 & 43 & 51 & 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 & 65 & 66 & 67 & 69 \\ 0 & 4 & 52 & 53 & 54 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 5 & 65 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 6 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 7 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 8 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 9 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 10 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 11 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 0 & 12 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \end{array}$$

Due to lack of computational resources this is an incomplete table.

Mathematica:

Clear[t];
nnn = 70
b = Table[
   t[n_, 1] = 1;
   t[1, k_] = 1; 
   t[n_, k_] := 
    t[n, k] = 
     If[n < k, 
      If[And[n > 1, k > 1], Sum[-t[k - i, n], {i, 1, n - 1}], 0], 
      If[And[n > 1, k > 1], Sum[-t[n - i, k], {i, 1, k - 1}], 0]];
   a = Table[Table[t[n, k]/n, {k, 1, nn}], {n, 1, nn}];
   Eigenvalues[a], {nn, 1, nnn}];
TableForm[
 Table[DeleteDuplicates[
   Flatten[Table[
     Table[If[b[[kk]][[1]] == Abs[b[[n]][[k]]], n, 0], {k, 1, n}], {n,
       1, nnn}]]], {kk, 1, 12}]]
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