An extension of a result of Erd\"os and Zaremba

TL;DR

This paper extends a classical result by Erd"os and Zaremba on the asymptotic behavior of a divisor sum function, introducing a new function and approach, with applications to inequalities involving gcd and divisor functions.

Contribution

The paper generalizes Erd"os and Zaremba's limit result to a new function involving logs and develops a novel proof technique, also improving related inequalities.

Findings

Established the asymptotic limit of the new function (n) involving logs.

Developed a new approach to prove the limit, different from previous methods.

Improved bounds on sums involving gcd and divisor functions.

Abstract

Erd\"os and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\g$, $\g$ being Euler's constant, where $\Phi(n)=\sum_{d|n} \frac{\log d}{d}$. We extend this result to the function $\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\g$. The proof requires to develop a new approach. As an application, we prove that for any $\eta>1$, any finite sequence of reals $\{c_k, k\in K\}$, $\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu) $, where $C(\eta)$ depends on $\eta$ only. This improves a recent result obtained by the author.