Mobius function

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1: 27.5 Inversion Formulas

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27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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27.5.3 $$g(n)=\sum _{d|n}f(d)⟺f(n)=\sum _{d|n}g(d)\mu \left(\frac{n}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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27.5.5 $$\ln n=\sum _{d|n}\mathrm{\Lambda }\left(d\right)⟺\mathrm{\Lambda }\left(n\right)=\sum _{d|n}(\ln d)\mu \left(\frac{n}{d}\right).$$

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Symbols:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function, $\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.5.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.5 and Ch.27

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27.5.8 $$g(n)=\prod _{d|n}f(d)⟺f(n)=\prod _{d|n}{\left(g\left(\frac{n}{d}\right)\right)}^{\mu \left(d\right)}.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.5 and Ch.27

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2: 27.17 Other Applications

§27.17 Other Applications

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3: 27.6 Divisor Sums

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27.6.2 $$\sum _{d|n}\mu \left(d\right)f(d)=\prod _{p|n}(1-f(p)),$$ $n>1$.

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. … ►

27.6.3 $$\sum _{d|n}|\mu \left(d\right)|=2^{\nu \left(n\right)},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\nu \left(n\right)$: number of distinct primes dividing a number, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.4 $$\sum _{d^2|n}\mu \left(d\right)=|\mu \left(n\right)|,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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27.6.7 $$\sum _{d|n}\mu \left(d\right){\left(\frac{n}{d}\right)}^k=J_k\left(n\right),$$

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Symbols:

$J_k\left(n\right)$: Jordan’s function, $\mu \left(n\right)$: Möbius function, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.6 and Ch.27

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4: 27.2 Functions

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27.2.12

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Defines:

$\mu \left(n\right)$: Möbius function

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a$: primitive root

A&S Ref:

24.3.1 I.A

Permalink:

http://dlmf.nist.gov/27.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►This is the Möbius function. …

5: 27.4 Euler Products and Dirichlet Series

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27.4.5 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)n^{-s}=\frac{1}{\zeta \left(s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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27.4.8 $$\sum _{n=1}^{\mathrm{\infty }}|\mu \left(n\right)|n^{-s}=\frac{\zeta \left(s\right)}{\zeta \left(2s\right)},$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mu \left(n\right)$: Möbius function, $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.4 and Ch.27

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6: 27.11 Asymptotic Formulas: Partial Sums

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27.11.12 $$\sum _{n\le x}\mu \left(n\right)=O\left(x{\mathrm{e}}^{-C\sqrt{\ln x}}\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mu \left(n\right)$: Möbius function, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $n$: positive integer, $x$: real number and $C$: constant

A&S Ref:

24.3.1 III

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E12

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.13 $$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}\sum _{n\le x}\mu \left(n\right)=0,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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27.11.14 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)}{n}=0,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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27.11.15 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)\ln n}{n}=-1.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E15

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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7: 27.7 Lambert Series as Generating Functions

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27.7.3 $$\sum _{n=1}^{\mathrm{\infty }}\mu \left(n\right)\frac{x^n}{1-x^n}=x,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 I.B

Permalink:

http://dlmf.nist.gov/27.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.7 and Ch.27

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8: 27.10 Periodic Number-Theoretic Functions

… ►It can also be expressed in terms of the Möbius function as a divisor sum: ►

27.10.5 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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9: Bibliography R

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On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.

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External Links:

Document, MathReview (M. A. Harrison), ZentralBlatt

Referenced by:

§27.5.

Encodings:

BibTeX

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10: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations …in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ 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 ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$. ►For arbitrary values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$, the general solutions of ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are transcendental, that is, they cannot be expressed in closed-form elementary functions. However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …