An extension of a result of Erdös and Zaremba

Abstract

Erdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{Φ(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{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 $\eta >1$, any finite sequence of reals $\{c_k,k\in K\}$, ${\sum }_{k,\ell \in K}{c}_k{c}_{\ell }\frac{\gcd (k,\ell {)}^2}{k\ell }\le C(\eta ){\sum }_{\nu \in K}{c}_{\nu }^2(\log \log \log \nu {)}^{\eta }\Psi (\nu )$, where $C(\eta )$ depends on $\eta$ only. This improves a recent result obtained by the author.

arXiv commentThis revised version has been recently published in Glasgow Mathematical Journal, Vol. 63, Issue 1, January 2021, pp. 193-222