Dirichlet convolution ($\ast$)

The Dirichlet convolution of arithmetic functions $f$ and $g$ isIt is associative and commutative, its identity is the function supported at $1$, and the Möbius function is the convolution inverse of the constant-one function.

The Dirichlet hyperbola method evaluates a summatory Dirichlet convolution by splitting the lattice points $ab\le x$ at $a,b\le \sqrt{x}$. For the divisor function,