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AN EXTENSION OF A RESULT OF ERDŐS AND ZAREMBA
Published online by Cambridge University Press: 13 May 2020
Abstract
Erdös and Zaremba showed that 
$ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$
, γ 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^\gamma$
. The proof requires 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(η) depends on η only. This improves a recent result obtained by the author.
- Type
- Research Article
- Information
- Copyright
- © Glasgow Mathematical Journal Trust 2020
References
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