Divisors on overlapped intervals and multiplicative functions

Abstract

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

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