Let $f, g : \Bbb{N} \to \Bbb{Z}$ be two arithmetic functions which are multiplicative (i.e. in terms of multiplicative on two coprime factors). Then what is: $$ h(n) = \sum_{d \mid n}f(d)g(d) $$ Note that we don't have $f(d)g(n/d)$ there which is the standard way (Dirichlet convolution) which is ...

Let all numbers be natural, or integer wherever appropriate. Let $(x\mid y) = \begin{cases}1 \text{ if } x \mid y \\ 0 \text{ if } x \nmid y \end{cases}$. Define. $$ f(n,m) = \sum_{d \mid n}(-1)^{\omega({d})}(d\mid m) $$ But $(-1)^{\omega(d)} = (-1)^{\omega(n)}(-1)^{\omega(n) + \omega(d)} = (-1)...

Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility character which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not divide. We define the non-divisibility character $(x\nmid y) = 1 - (x|y)$. Clearly, it equals $1...