Sqrt[MoebiusMu[k]*k]

(*start*)
nn = 200;
s = N[8, 20];
(Sum[Sum[If[Mod[n, k] == 0, (MoebiusMu[k]*k)^(1/2)/n^s, 0], {k, 1,
     nn}], {n, 1, nn}])^2
(A = Table[
    Table[If[Mod[n, k] == 0, (MoebiusMu[k]*k)^(1/2)/n^s, 0], {k, 1,
      nn}], {n, 1, nn}]);
N[Total[Total[A.Transpose[A]]], 20]
Zeta[s]*Zeta[s]/Zeta[2*s - 1]
(*end*)

Out[446]= 1.00814049998960894694 + 0.01168365002895327553 I

Out[448]= 1.0081404999896092120

Out[449]= 1.008140499989609234


(*start*)
nn = 200;
s = N[8, 20];
(Sum[Sum[If[Mod[n, k] == 0, (MoebiusMu[k])^(1/2)/n^s, 0], {k, 1,
     nn}], {n, 1, nn}])^2
(A = Table[
    Table[If[Mod[n, k] == 0, (MoebiusMu[k])^(1/2)/n^s, 0], {k, 1,
      nn}], {n, 1, nn}]);
N[Total[Total[A.Transpose[A]]], 20]
Zeta[s]*Zeta[s]/Zeta[2*s]
(*end*)

Out[656]= 1.00815593032900188265 + 0.008189189928767649031 I

Out[658]= 1.0081559303290019133

Out[659]= 1.008155930329001935