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Questions tagged [dirichlet-convolution]

Use this tag for questions related to Dirichlet convolution in number theory

Dirichlet convolution is a type of convolution used in number theory for arithmetic functions. It forms a commutative ring under pointwise addition. It is defined as

\begin{equation*} (f\ast g)(n)=\sum_{d|n}f(d)g(\frac{n}{d}) \end{equation*}

where $f$ and $g$ are two arithmetic functions, and $d$ is a divisor of $n.$

Note that the multiplication of Dirichlet series is compatible with Dirichlet convolution.

121 questions
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Dirichlet convolution k times.

We know that $-\sum\limits_{d|n}\mu(d)\log d=\Lambda(n)$. Using this we can obtain $$(\Lambda*\Lambda)(n)=\Lambda(n)\log n+\sum\limits_{d|n}\mu(d)\log^2d.$$ In general if I write Dirichlet convolution $k$ times $$A_k=\Lambda*\Lambda*.....*\Lambda$$,…
SSK
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Dirichlet Convolution Identities

I am unsure of the proof of the identity involving the identity arithmetic function, $e$. That is, the identity $f\ast e=f$. My proof so far is: $$[f\ast e](n)=\sum\limits_{d\mid n}f(d)e(\frac{n}{d})$$ Letting $d=n$ to make…
ceols
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