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27 Functions of Number TheoryMultiplicative Number Theory

§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)=d|nf(d)g(nd),

called the Dirichlet product (or convolution) of f and g. The set of all number-theoretic functions f with f(1)0 forms an abelian group under Dirichlet multiplication, with the function 1/n 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 ζ(s)(1/ζ(s))=1 is equivalent to the identity

27.5.2 d|nμ(d)=1n,

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

27.5.3 g(n)=d|nf(d)f(n)=d|ng(d)μ(nd).

Special cases of Möbius inversion pairs are:

27.5.4 n=d|nϕ(d)ϕ(n)=d|ndμ(nd),
27.5.5 lnn=d|nΛ(d)Λ(n)=d|n(lnd)μ(nd).

Other types of Möbius inversion formulas include:

27.5.6 G(x)=nxF(xn)F(x)=nxμ(n)G(xn),
27.5.7 G(x)=m=1F(mx)msF(x)=m=1μ(m)G(mx)ms,
27.5.8 g(n)=d|nf(d)f(n)=d|n(g(nd))μ(d).

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