Editing Square-free integer (section)

=== Computation of {{Math|''Q''(''x'')}} ===
The squarefree integers {{Math|&leq; ''x''}} can be identified and counted in {{Math|[[big O notation|''Õ'']](''x'')}} time by using a modified [[Sieve of Eratosthenes]].  If only {{Math|''Q''(''x'')}} is desired, and not a list of the numbers that it counts, then ({{EquationNote|1}}) can be used to compute {{Math|''Q''(''x'')}} in {{Math|''Õ''({{radical|''x''}})}} time.  The largest known value of {{Math|''Q''(''x'')}}, for {{Math|1=''x'' = 10<sup>36</sup>}}, was computed by Jakub Pawlewicz in 2011 using an algorithm that achieves {{Math|''Õ''(''x''<sup>2/5</sup>)}} time,<ref>{{cite arXiv |eprint=1107.4890 |last1=Pawlewicz |first1=Jakub |title=Counting Square-Free Numbers |date=2011 |class=math.NT }}</ref> and an algorithm taking {{Math|''Õ''(''x''<sup>1/3</sup>)}} time has been outlined but not implemented.{{r|HirschKesslerMendlovic2024|at=&sect;5.5}}