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

Erd"os and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\g$,$\g$beingEule{r}^{\mathrm{\prime }}sconstant,where$\Phi(n)=\sum_{d|n} \frac{\log d}{d}$\mathrm{.}Weextendthisresulttothefunction$\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)$dependson$\eta$ only. This improves a recent result obtained by the author.