An extension of a result of Erdös and Zaremba
Erdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{Φ(n)}{(\log\log n)^2}=e^\g$, $\g$ being Euler's constant, where $Φ(n)=\sum_{d|n} \frac{\log d}{d}$. We extend this result to the function $Ψ(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{Ψ(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 $η>1$, 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(η) \sum_{ν\in K} c_ν^2(\log\log\log ν)^ηΨ(ν) $, where $C(η)$ depends on $η$ only. This improves a recent result obtained by the author.
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