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Note: Table code included as requested:

a = 7;
x = 10^a;
kk = 20;
TableForm[Transpose[{Table[x, {x, 1, kk}], Table[Round[landau[k, x]], {k, 1, kk}], 
Table[Round[cdf[k, x]], {k, 1, kk}], Table[actual[k, x], {k, 1, kk}]}], 
TableHeadings -> {None, {"k  ", "Landau", "CDF   ", "Actual"}}, 
TableSpacing -> {2, 3, 0}]

As not necessarily provable results were asked for, I have found the following quite useful:

As not necessarily proven results were asked for, I have found the following quite accurate: