({{, (RefLink[DirichletConvolve, paclet:ref/DirichletConvolve])[f,g,n,m] 
gives the Dirichlet convolution of the expressions f and g. }})

OpenersNotesSection(Spacer1)Details

The Dirichlet convolution (f *g)(m) of two functions f(n) and g(n) is given by (∑_(d∣m))f(d) g((m/d)).

If m is positive, (RefLink[DirichletConvolve, paclet:ref/DirichletConvolve])[f,g,n,m] can always be evaluated as a finite sum.

OpenersPrimaryExamplesSection(Spacer1)Examples  (3)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

DirichletConvolve[n,n,n,m]

OpenersExampleSection1(Spacer1)Scope  (1)

Polynomial functions:

DirichletConvolve[n+2,n^2+1,n,m]

OpenersExampleSection3(Spacer1)Properties & Relations  (1)

(RefLink[KroneckerDelta, paclet:ref/KroneckerDelta]) is an identity element:

DirichletConvolve[f[n],KroneckerDelta[n-1],n,m]

Symmetry:

{DirichletConvolve[n,n^2,n,m],DirichletConvolve[n^2,n,n,m]}

Distributivity with respect to the addition:

DirichletConvolve[f[n],g[n]+h[n],n,m]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[DivisorSum, paclet:ref/DivisorSum])  ▪  (RefLink[DirichletTransform, paclet:ref/DirichletTransform])  ▪  (RefLink[DiscreteConvolve, paclet:ref/DiscreteConvolve])  ▪  (RefLink[MoebiusMu, paclet:ref/MoebiusMu])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Multiplicative Number Theory, paclet:guide/MultiplicativeNumberTheory])}, {▪(RefLinkPlain[Analytic Number Theory, paclet:guide/AnalyticNumberTheory])}, {▪(RefLinkPlain[Additive Number Theory, paclet:guide/AdditiveNumberTheory])}, {▪(RefLinkPlain[Summation Transforms, paclet:guide/SummationTransforms])}})}})