# §27.5 Inversion Formulas

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

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 $\mathop{\zeta\/}\nolimits\!\left(s\right)\cdot(\ifrac{1}{\mathop{\zeta\/}% \nolimits\!\left(s\right)})=1$ is equivalent to the identity

 27.5.2 $\sum_{d\mathbin{|}n}\mathop{\mu\/}\nolimits\!\left(d\right)=\left\lfloor\frac{% 1}{n}\right\rfloor,$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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

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

 27.5.3 $g(n)=\sum_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\sum_{d\mathbin{|}n}g(d)% \mathop{\mu\/}\nolimits\!\left(\frac{n}{d}\right).$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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

Special cases of Möbius inversion pairs are:

 27.5.4 $n=\sum_{d\mathbin{|}n}\mathop{\phi\/}\nolimits\!\left(d\right)% \Longleftrightarrow\mathop{\phi\/}\nolimits\!\left(n\right)=\sum_{d\mathbin{|}% n}d\mathop{\mu\/}\nolimits\!\left(\frac{n}{d}\right),$ Symbols: $\mathop{\phi\/}\nolimits\!\left(\NVar{n}\right)$: Euler’s totient, $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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
 27.5.5 $\mathop{\ln\/}\nolimits n=\sum_{d\mathbin{|}n}\mathop{\Lambda\/}\nolimits\!% \left(d\right)\Longleftrightarrow\mathop{\Lambda\/}\nolimits\!\left(n\right)=% \sum_{d\mathbin{|}n}(\mathop{\ln\/}\nolimits d)\mathop{\mu\/}\nolimits\!\left(% \frac{n}{d}\right).$ Symbols: $\mathop{\Lambda\/}\nolimits\!\left(\NVar{n}\right)$: Mangoldt’s function, $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: Möbius function, $\mathop{\ln\/}\nolimits\NVar{z}$: 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 $\mathop{\ln\/}\nolimits$ from $\mathrm{log}$. Reported 2015-06-08 See also: Annotations for 27.5

Other types of Möbius inversion formulas include:

 27.5.6 $G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)\Longleftrightarrow F(x)=\sum_{n% \leq x}\mathop{\mu\/}\nolimits\!\left(n\right)G\left(\frac{x}{n}\right),$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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
 27.5.7 $G(x)=\sum_{m=1}^{\infty}\frac{F(mx)}{m^{s}}\Longleftrightarrow F(x)=\sum_{m=1}% ^{\infty}\mathop{\mu\/}\nolimits\!\left(m\right)\frac{G(mx)}{m^{s}},$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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
 27.5.8 $g(n)=\prod_{d\mathbin{|}n}f(d)\Longleftrightarrow f(n)=\prod_{d\mathbin{|}n}% \left(g\left(\frac{n}{d}\right)\right)^{\mathop{\mu\/}\nolimits\!\left(d\right% )}.$ Symbols: $\mathop{\mu\/}\nolimits\!\left(\NVar{n}\right)$: 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

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