Divisor Counting Function as Convolution
Theorem
The divisor counting function can be expressed as the Dirichlet convolution
\[ d = 1 \ast 1\]
where \(1\) is the constant function \(1(n) = 1\).
Proof
\[ (1 \ast 1)(n) = \sum_{d \mid n} 1(d) \cdot 1\left(\frac{d}{n}\right) = \sum_{d \mid n} 1 = d(n)\]