Divisors on overlapped intervals and multiplicative functions

Abstract

Consider the real numbers $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $\mathcal{L}_{n,k} = \left]\ell_{n,k}-\ln 3,\ell_{n,k}\right]$. For all $n \geq 1$, define $$ \frac{L_n(q)}{q^{n-1}} = \sum_{d|n}\sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \,q^k, $$ where $\boldsymbol{1}_{A}(x)$ is the characteristic function of the set $A$. Let $σ(n)$ be sum of divisors of $n$. We will prove that $\textbf{A002324}(n) = 4\,σ(n) - 3\,L_n(1)$ and $\textbf{A096936}(n) = L_n(-1)$, which are well-known multiplicative functions related to the number of representations of $n$ by a given quadratic form.