Calculation
Neither of the methods mentionned previously leads to practical algorithms to calculate the Mertens function. Using sieve methods similar to those used in prime counting, the Mertens function has been computed for an increasing range of n.
| Person | Year | Limit |
| Mertens | 1897 | 104 |
| von Sterneck | 1897 | 1.5×105 |
| von Sterneck | 1901 | 5×105 |
| von Sterneck | 1912 | 5×106 |
| Neubauer | 1963 | 108 |
| Cohen and Dress | 1979 | 7.8×109 |
| Dress | 1993 | 1012 |
| Lioen and van de Lune | 1994 | 1013 |
| Kotnik and van de Lune | 2003 | 1014 |
The Mertens function for all integer values up to N may be computed in O(N2/3+ε) time, while better methods are known. Elementary algorithms exist to compute isolated values of M(N) in O(N2/3*(ln ln(N))1/3) time.
See A084237 for values of M(N) at powers of 10.
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