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Mobius function

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1: 27.5 Inversion Formulas
27.5.2 d | n μ ( d ) = 1 n ,
27.5.3 g ( n ) = d | n f ( d ) f ( n ) = d | n g ( d ) μ ( n d ) .
27.5.5 ln n = d | n Λ ( d ) Λ ( n ) = d | n ( ln d ) μ ( n d ) .
27.5.8 g ( n ) = d | n f ( d ) f ( n ) = d | n ( g ( n d ) ) μ ( d ) .
2: 27.17 Other Applications
§27.17 Other Applications
3: 27.6 Divisor Sums
27.6.2 d | n μ ( d ) f ( d ) = p | n ( 1 - f ( p ) ) , n > 1 .
Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …
27.6.3 d | n | μ ( d ) | = 2 ν ( n ) ,
27.6.4 d 2 | n μ ( d ) = | μ ( n ) | ,
27.6.7 d | n μ ( d ) ( n d ) k = J k ( n ) ,
4: 27.2 Functions
27.2.12 μ ( n ) = { 1 , n = 1 , ( - 1 ) ν ( n ) , a 1 = a 2 = = a ν ( n ) = 1 , 0 , otherwise .
This is the Möbius function. …
5: 27.4 Euler Products and Dirichlet Series
27.4.5 n = 1 μ ( n ) n - s = 1 ζ ( s ) , s > 1 ,
27.4.8 n = 1 | μ ( n ) | n - s = ζ ( s ) ζ ( 2 s ) , s > 1 ,
6: 27.11 Asymptotic Formulas: Partial Sums
27.11.12 n x μ ( n ) = O ( x e - C ln x ) , x ,
27.11.13 lim x 1 x n x μ ( n ) = 0 ,
27.11.14 lim x n x μ ( n ) n = 0 ,
27.11.15 lim x n x μ ( n ) ln n n = - 1 .
7: 27.7 Lambert Series as Generating Functions
27.7.3 n = 1 μ ( n ) x n 1 - x n = x ,
8: 27.10 Periodic Number-Theoretic Functions
It can also be expressed in terms of the Möbius function as a divisor sum:
9: Bibliography R
  • G. Rota (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.
  • 10: 32.2 Differential Equations
    They are distinct modulo Möbius (bilinear) transformations …in which a ( z ) , b ( z ) , c ( z ) , d ( z ) , and ϕ ( z ) are locally analytic functions. The fifty equations can be reduced to linear equations, solved in terms of elliptic functions (Chapters 22 and 23), or reduced to one of P I P VI . For arbitrary values of the parameters α , β , γ , and δ , the general solutions of P I P VI  are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations P II P VI  have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …