Mobius inversion formulas

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11: 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:

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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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12: 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:

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See also:

Annotations for §27.7 and Ch.27

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13: 1.9 Calculus of a Complex Variable

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1.9.6 $$\omega =\arctan \left(\left|y/x\right|\right)\in [0,\frac{1}{2}\pi ].$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\arctan z$: arctangent function and $\left|x\right|$: absolute value of $x$

A&S Ref:

3.7.4

Permalink:

http://dlmf.nist.gov/1.9.E6

Encodings:

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Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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Cauchy’s Integral Formula

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Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. … ►

§1.9(vii) Inversion of Limits

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14: 12.16 Mathematical Applications

… ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …

15: 35.2 Laplace Transform

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Inversion Formula

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16: 24.5 Recurrence Relations

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§24.5(iii) Inversion Formulas

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17: 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:

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See also:

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

►This is the Möbius function. …

18: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

19: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. … ►In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have …

20: 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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