## Mertens function

Mathematica

``` nn = 1000 Monitor[aa = Table[Sum[MoebiusMu[k]*Floor[n/k]^(0), {k, 1, n}], {n, 1, nn}];, n] Monitor[bb = Table[Sum[MoebiusMu[k]*Floor[n/k]^(1/2), {k, 1, n}], {n, 1, nn}];, n + 1000] Monitor[cc = Table[(6/Pi^2)*n^(1/2), {n, 1, nn}];, n + 2000] ListLinePlot[{aa, bb, -cc, bb + 2*cc - 2*cc[[1]], cc}, ImageSize -> Full]```

``` Print["These are equal:"] Clear[t]; nn = 12; rowsumexponent = 1/2; t[n_, k_] := t[n, k] = If[n = k, t[Floor[n/k], 1]]]]]; MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]; gg = Table[t[n, 1], {n, 1, 12}]; dd = Table[ Sum[MoebiusMu[k]*Floor[n/k]^(rowsumexponent), {k, 1, n}], {n, 1, nn}]; MatrixForm[Transpose[{gg, dd, dd - gg}]] ```

```Print["But unfortunately these are not equal:"] Clear[t]; nn = 12; rowsumexponent = 1/2; t[n_, k_] := t[n, k] = If[n = k, t[Floor[n/k], 1]]]]]; MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]; gg = Table[t[n, 1], {n, 1, 12}]; dd = (6/Pi^2)* Table[Sum[MoebiusMu[k]*Floor[n/k]^(rowsumexponent), {k, 1, n}], {n, 1, nn}]; MatrixForm[Transpose[{gg, dd, dd - gg}]] ```