Reuns von Mangoldt function convergence Riemann hypothesis

(*Mathematica start*)
Monitor[a1 =
  Table[-1/n^(1/2)/Log[n]^3 +
    Sum[If[a == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}] -
    Sum[Sum[If[a*b == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}], {b,
      2, n}] +
    Sum[Sum[Sum[
       If[a*b*c == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2, n}], {b, 2,
       n}], {c, 2, n}] -
    Sum[Sum[Sum[
       Sum[If[a*b*c*d == n, Log[a]/n^(1/2)/Log[n]^3, 0], {a, 2,
         n}], {b, 2, n}], {c, 2, n}], {d, 2, n}], {n, 2, 2^5 - 1}], n]
a2 = Table[(MangoldtLambda[n] - 1)/n^(1/2)/Log[n]^3, {n, 2, 2^5 - 1}]
N[a1 - a2]
Chop[%]
(*end*)


(*Mathematica start*)
nn = 2^7;
A = Table[
   Table[If[Mod[n, k] == 0, ""[n], 0]*If[n > k, 1, 0], {k, 1,
     nn}], {n, 1, nn}];
Expand[(MatrixPower[A, 0])[[2^0 ;; 2^1 - 1, 1]]]
Expand[(MatrixPower[A, 1])[[2^1 ;; 2^2 - 1, 1]]]
Expand[(MatrixPower[A, 2])[[2^2 ;; 2^3 - 1, 1]]]
Expand[(MatrixPower[A, 3])[[2^3 ;; 2^4 - 1, 1]]]
Expand[(MatrixPower[A, 4])[[2^4 ;; 2^5 - 1, 1]]]
Expand[(MatrixPower[A, 5])[[2^5 ;; 2^6 - 1, 1]]]
Expand[(MatrixPower[A, 6])[[2^6 ;; 2^7 - 1, 1]]]
(*end*)

(*****14.06.2021 klockan 23:54 **********************************************)

Limit[1/Sqrt[2^(n - 1)]/Log[2^(n - 1)]^3*
   Log[2^(n - 1)]/(n - 1) + (n - 1)/Sqrt[3*2^(n - 1 - 1)]/
    Log[3*2^(n - 1 - 1)]^3*Log[3*2^(n - 1 - 1)]/(n - 1), n -> Infinity]

0

Limit[1/Sqrt[2^(n - 1)]/Log[2^(n - 1)]^3*
   Log[2^(n - 1)]/(n - 1) + (n - 1)/Sqrt[3*2^(n - 1 - 1)]/
    Log[3*2^(n - 1 - 1)]^3*
   Log[3*2^(n - 1 - 1)]/(n - 1)/(
     1/Sqrt[2^(n + 1 - 1)]/Log[2^(n + 1 - 1)]^3*
       Log[2^(n + 1 - 1)]/(n + 1 - 1) + (n + 1 - 1)/
         Sqrt[3*2^(n + 1 - 1 - 1)]/Log[3*2^(n + 1 - 1 - 1)]^3*
       Log[3*2^(n + 1 - 1 - 1)]/(n + 1 - 1)          ), n -> Infinity]

Sqrt[2]

(***** 14.06.2021 klockan 23:54 **********************************************)


(* https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown *)