= Multiplicativity of Dirichlet convolution
{c}
{title2=$f,g\ \text{multiplicative}\ \Longrightarrow\ f*g\ \text{multiplicative}$}
The <Dirichlet convolution> of two <multiplicative functions> is a <multiplicative function>. For positive <coprime integers> $m,n$, every positive <integer divisor> of $mn$ has a unique form $de$ with $d\mid m$ and $e\mid n$, by <prime factorization>. The pairs $d,e$ and $m/d,n/e$ are each <coprime integers>, so the defining rules for the two <multiplicative functions> separate the sum:
$$
(f*g)(mn)=\sum_{d\mid m}\sum_{e\mid n}f(d)f(e)g(m/d)g(n/e)=(f*g)(m)(f*g)(n).
$$
Also $(f*g)(1)=f(1)g(1)=1$. This short argument proves <multiplicativity> of both the <divisor function> $\mathbf1*\mathbf1$ and the <sum-of-divisors function> $\operatorname{id}*\mathbf1$. The result need not be a <completely multiplicative function>, even when both inputs are: $\mathbf1*\mathbf1=\tau$ fails at $2\cdot2$. https://darijgrinberg.gitlab.io/floor.pdf[Darij Grinberg, MIT 18.781 lecture notes, Theorem 2.6.1]{ref}
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