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table of values of the Möbius function and the Mertens function
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(Data Structure)
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The following table lists the values of the Möbius function $\mu(n)$ and the Mertens function $M(n)$ for $0 < n < 101$ The Möbius function is defined as $\mu(n) = (-1)^{\omega(n)}$ (where $\omega(n)$ is the number of distinct prime factors function) for squarefree numbers, and $\mu(n) = 0$ for any integer with a repeated prime factor. The Mertens function is the matching summatory function for the Möbius function, $$M(n) = \sum_{i = 1}^n \mu(i).$$
| $n$ |
$\mu(n)$ |
$M(n)$ |
$n$ |
$\mu(n)$ |
$M(n)$ |
$n$ |
$\mu(n)$ |
$M(n)$ |
$n$ |
$\mu(n)$ |
$M(n)$ |
| 1 |
1 |
1 |
26 |
1 |
$-1$ |
51 |
1 |
$-2$ |
76 |
0 |
$-3$ |
| 2 |
$-1$ |
0 |
27 |
0 |
$-1$ |
52 |
0 |
$-2$ |
77 |
1 |
$-2$ |
| 3 |
$-1$ |
$-1$ |
28 |
0 |
$-1$ |
53 |
$-1$ |
$-3$ |
78 |
$-1$ |
$-3$ |
| 4 |
0 |
$-1$ |
29 |
$-1$ |
$-2$ |
54 |
0 |
$-3$ |
79 |
$-1$ |
$-4$ |
| 5 |
$-1$ |
$-2$ |
30 |
$-1$ |
$-3$ |
55 |
1 |
$-2$ |
80 |
0 |
$-4$ |
| 6 |
1 |
$-1$ |
31 |
$-1$ |
$-4$ |
56 |
0 |
$-2$ |
81 |
0 |
$-4$ |
| 7 |
$-1$ |
$-2$ |
32 |
0 |
$-4$ |
57 |
1 |
$-1$ |
82 |
1 |
$-3$ |
| 8 |
0 |
$-2$ |
33 |
1 |
$-3$ |
58 |
1 |
0 |
83 |
$-1$ |
$-4$ |
| 9 |
0 |
$-2$ |
34 |
1 |
$-2$ |
59 |
$-1$ |
$-1$ |
84 |
0 |
$-4$ |
| 10 |
1 |
$-1$ |
35 |
1 |
$-1$ |
60 |
0 |
$-1$ |
85 |
1 |
$-3$ |
| 11 |
$-1$ |
$-2$ |
36 |
0 |
$-1$ |
61 |
$-1$ |
$-2$ |
86 |
1 |
$-2$ |
| 12 |
0 |
$-2$ |
37 |
$-1$ |
$-2$ |
62 |
1 |
$-1$ |
87 |
1 |
$-1$ |
| 13 |
$-1$ |
$-3$ |
38 |
1 |
$-1$ |
63 |
0 |
$-1$ |
88 |
0 |
$-1$ |
| 14 |
1 |
$-2$ |
39 |
1 |
0 |
64 |
0 |
$-1$ |
89 |
$-1$ |
$-2$ |
| 15 |
1 |
$-1$ |
40 |
0 |
0 |
65 |
1 |
0 |
90 |
0 |
$-2$ |
| 16 |
0 |
$-1$ |
41 |
$-1$ |
$-1$ |
66 |
$-1$ |
$-1$ |
91 |
1 |
$-1$ |
| 17 |
$-1$ |
$-2$ |
42 |
$-1$ |
$-2$ |
67 |
$-1$ |
$-2$ |
92 |
0 |
$-1$ |
| 18 |
0 |
$-2$ |
43 |
$-1$ |
$-3$ |
68 |
0 |
$-2$ |
93 |
1 |
0 |
| 19 |
$-1$ |
$-3$ |
44 |
0 |
$-3$ |
69 |
1 |
$-1$ |
94 |
1 |
1 |
| 20 |
0 |
$-3$ |
45 |
0 |
$-3$ |
70 |
$-1$ |
$-2$ |
95 |
1 |
2 |
| 21 |
1 |
$-2$ |
46 |
1 |
$-2$ |
71 |
$-1$ |
$-3$ |
96 |
0 |
2 |
| 22 |
1 |
$-1$ |
47 |
$-1$ |
$-3$ |
72 |
0 |
$-3$ |
97 |
$-1$ |
1 |
| 23 |
$-1$ |
$-2$ |
48 |
0 |
$-3$ |
73 |
$-1$ |
$-4$ |
98 |
0 |
1 |
| 24 |
0 |
$-2$ |
49 |
0 |
$-3$ |
74 |
1 |
$-3$ |
99 |
0 |
1 |
| 25 |
0 |
$-2$ |
50 |
0 |
$-3$ |
75 |
0 |
$-3$ |
100 |
0 |
1 |
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"table of values of the Möbius function and the Mertens function" is owned by PrimeFan.
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Cross-references: function, matching, prime factor, integer, numbers, squarefree, number of distinct prime factors function, Mertens function, Möbius function
There is 1 reference to this entry.
This is version 2 of table of values of the Möbius function and the Mertens function, born on 2008-06-03, modified 2008-06-04.
Object id is 10652, canonical name is TableOfValuesOfTheMobiusFunctionAndTheMertensFunction.
Accessed 492 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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