27 Functions of Number Theory Multiplicative Number Theory 27.4 Euler Products and Dirichlet Series 27.6 Divisor Sums

§27.5 Inversion Formulas

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Keywords:: Dirichlet product (or convolution), Möbius inversion, Möbius inversion formulas, inversion formulas, number theory, number-theoretic functions, pairs
Notes:: See Apostol (1976, Chapter 2 and p. 228). For (27.5.7) use (27.5.2) and formal substitution.
Permalink:: http://dlmf.nist.gov/27.5
See also:: Annotations for Ch.27

If a Dirichlet series $F(s)$ generates $f(n)$ , and $G(s)$ generates $g(n)$ , then the product $F(s)G(s)$ generates

27.5.1		$h(n)=\sum_{d\mathbin{\|}n}f(d)g\left(\frac{n}{d}\right),$
ⓘ Symbols: $d$ : positive integer, $n$ : positive integer, $f$ : function, $g$ : function and $h$ : function Permalink: http://dlmf.nist.gov/27.5.E1 Encodings: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

called the Dirichlet product (or convolution) of $f$ and $g$ . The set of all number-theoretic functions $f$ with $f(1)\neq 0$ forms an abelian group under Dirichlet multiplication, with the function $\left\lfloor 1/n\right\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). The multiplicative functions are a subgroup of this group. Generating functions yield many relations connecting number-theoretic functions. For example, the equation $\zeta\left(s\right)\cdot(\ifrac{1}{\zeta\left(s\right)})=1$ is equivalent to the identity

27.5.2		$\sum_{d\mathbin{\|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$
ⓘ Symbols: $\mu\left(\NVar{n}\right)$ : Möbius function, $\left\lfloor\NVar{x}\right\rfloor$ : 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: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

which, in turn, is the basis for the Möbius inversion formula relating sums over divisors:

27.5.3		$g(n)=\sum_{d\mathbin{\|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{\|}n}g(d)% \mu\left(\frac{n}{d}\right).$
ⓘ Symbols: $\mu\left(\NVar{n}\right)$ : 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: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

Special cases of Möbius inversion pairs are:

27.5.4		$n=\sum_{d\mathbin{\|}n}\phi\left(d\right)\Longleftrightarrow\phi\left(n\right)=% \sum_{d\mathbin{\|}n}d\mu\left(\frac{n}{d}\right),$
ⓘ Symbols: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\mu\left(\NVar{n}\right)$ : Möbius function, $d$ : positive integer and $n$ : positive integer A&S Ref: 24.3.2 II.B Permalink: http://dlmf.nist.gov/27.5.E4 Encodings: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

27.5.5		$\ln n=\sum_{d\mathbin{\|}n}\Lambda\left(d\right)\Longleftrightarrow\Lambda\left% (n\right)=\sum_{d\mathbin{\|}n}(\ln d)\mu\left(\frac{n}{d}\right).$
ⓘ Symbols: $\Lambda\left(\NVar{n}\right)$ : Mangoldt’s function, $\mu\left(\NVar{n}\right)$ : Möbius function, $\ln\NVar{z}$ : principal branch of logarithm function, $d$ : positive integer and $n$ : positive integer Permalink: http://dlmf.nist.gov/27.5.E5 Encodings: TeX, pMML, png Change of Notation (effective with 1.0.10): The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ . See also: Annotations for §27.5 and Ch.27

Other types of Möbius inversion formulas include:

27.5.6		$G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)\Longleftrightarrow F(x)=\sum_{n% \leq x}\mu\left(n\right)G\left(\frac{x}{n}\right),$
ⓘ Symbols: $\mu\left(\NVar{n}\right)$ : Möbius function, $n$ : positive integer, $x$ : real number, $F(s)$ : Dirichlet series and $G(s)$ : Dirichlet series A&S Ref: 24.3.1 II.C Permalink: http://dlmf.nist.gov/27.5.E6 Encodings: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

27.5.7		$G(x)=\sum_{m=1}^{\infty}\frac{F(mx)}{m^{s}}\Longleftrightarrow F(x)=\sum_{m=1}% ^{\infty}\mu\left(m\right)\frac{G(mx)}{m^{s}},$
ⓘ Symbols: $\mu\left(\NVar{n}\right)$ : Möbius function, $m$ : positive integer, $x$ : real number, $F(s)$ : Dirichlet series and $G(s)$ : Dirichlet series Referenced by: §27.17, §27.5 Permalink: http://dlmf.nist.gov/27.5.E7 Encodings: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

27.5.8		$g(n)=\prod_{d\mathbin{\|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{\|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.$
ⓘ Symbols: $\mu\left(\NVar{n}\right)$ : 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: TeX, pMML, png See also: Annotations for §27.5 and Ch.27

For a general theory of Möbius inversion with applications to combinatorial theory see Rota (1964).

27.4 Euler Products and Dirichlet Series 27.6 Divisor Sums

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§27.5 Inversion Formulas

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