Divisors on overlapped intervals and multiplicative functions

TL;DR

This paper investigates the properties of certain multiplicative functions related to divisor sums and quadratic form representations, using intervals defined via logarithmic functions and their overlaps.

Contribution

It establishes explicit formulas connecting well-known multiplicative functions to sums over logarithmic interval overlaps, providing new insights into their structure.

Findings

A002324(n) = 4σ(n) - 3L_n(1)

A096936(n) = L_n(-1)

Connections between divisor functions and interval overlaps

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 $\sigma(n)$ be sum of divisors of $n$. We will prove that $\textbf{A002324}(n) = 4\,\sigma(n) - 3\,L_n(1)$ 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.