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 generates , and generates , then the product generates

27.5.1 $h(n)=\sum_{d\mathbin{|}n}f(d)g\left(\frac{n}{d}\right),$

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Symbols:: : positive integer, : positive integer, : function, : function and : 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 and . The set of all number-theoretic functions 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 $\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, $\left\lfloor\NVar{x}\right\rfloor$ : floor of , : positive integer and : 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 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mu\left(\frac{n}{d}\right).$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : positive integer, : function and : 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 $n=\sum_{d\mathbin{|}n}\phi\left(d\right)\Longleftrightarrow\phi\left(n\right)=% \sum_{d\mathbin{|}n}d\mu\left(\frac{n}{d}\right),$

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Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : 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 $\ln n=\sum_{d\mathbin{|}n}\Lambda\left(d\right)\Longleftrightarrow\Lambda\left% (n\right)=\sum_{d\mathbin{|}n}(\ln d)\mu\left(\frac{n}{d}\right).$

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Symbols:: $\Lambda\left(\NVar{n}\right)$ : Mangoldt’s function, $\mu\left(\NVar{n}\right)$ : Möbius function, $\ln\NVar{z}$ : principal branch of logarithm function, : positive integer and : 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 $G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)\Longleftrightarrow F(x)=\sum_{n% \leq x}\mu\left(n\right)G\left(\frac{x}{n}\right),$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : real number, : Dirichlet series and : 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 $G(x)=\sum_{m=1}^{\infty}\frac{F(mx)}{m^{s}}\Longleftrightarrow F(x)=\sum_{m=1}% ^{\infty}\mu\left(m\right)\frac{G(mx)}{m^{s}},$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : real number, : Dirichlet series and : 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 $g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mu\left(d\right)}.$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer, : positive integer, : function and : 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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