Dirichlet Series of Convolution of Arithmetic Functions

Theorem

Let $f, g: \N \to \C$ be arithmetic functions.

Let $h = f * g$ be their Dirichlet convolution.

Let $F, G, H$ be their Dirichlet series.

Let $s$ be a complex number such that $\map F s$ and $\map G s$ converge absolutely.

Then $\map H s$ converges absolutely to $\map F s \times \map G s$.

Proof

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Also see

• Dirichlet Series of Inverse of Arithmetic Function
• Upper Bound for Abscissa of Absolute Convergence of Product of Dirichlet Series