determining a convolution of an arithmetic function

Question

Let be $ \lambda: \mathbb{N} \rightarrow \mathbb{C}$ be an arithmetic function $$ \lambda (n) := (-1)^{e_1+\dots+e_r} $$ where $p_1^{e_1}...p_r^{e_r} $ is the prime factorization of $n$ and it is $ \lambda (1)=1$.

My question is how can I determine the convolution $ \lambda \ast \lambda$ ?

The convolution of two arithmetic fuctions is given by

$$( \phi \ast \lambda)(n) := \sum_{d | n} \phi(d) \lambda (n | d) $$

Would appreciate any help!

Answer 1

The Liouville function $\lambda$ is completely multiplicative, i.e. \begin{align*} &\lambda(1)=1\\ &\lambda(m\cdot n)=\lambda(m)\lambda(n)\qquad m,n\in\mathbb{N}\tag{1} \end{align*}

We obtain for $n\in\mathbb{N}$: \begin{align*} \color{blue}{\left(\lambda\ast\lambda\right)(n)}&=\sum_{d\mid n}\lambda(d)\lambda\left(\frac{n}{d}\right)\\ &=\sum_{d\mid n}\lambda(n)\tag{2}\\ &=\lambda(n)\sum_{d\mid n}1\\ &\,\,\color{blue}{=\lambda(n)\tau(n)}\tag{3} \end{align*}

Comment:

• In (2) we use the complete multiplicative property of $\lambda$.

• In (3) we use the divisor function $\tau$, the number of positive divisors of $n$.