The convolution of two functions can mean various things.

In number theoretic context, convolution of two functions $f,g : \mathbb{N} \rightarrow \mathbb{C}$ usually means Dirichlet convolution, defined as $\displaystyle f * g = \sum_{d\mid n} f(d)g\left( \frac{n}{d} \right)$.

In analytic context, convolution of functions $\displaystyle f, g$ usually means a function of the form $\int f(\tau) g(t-\tau) d\tau$.

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