Large values of and \sigma (n)/n
Volume 209 / 2023
Abstract
Let n be a positive integer, \varphi (n) the Euler totient function, and \sigma (n)=\sum _{d\mid n}d the sum of the divisors of n. It is easy to prove that \sigma (n)/n\le n/\varphi (n). Landau proved that when n\to \infty , \limsup n/(\varphi (n)\log \log n) = e^\gamma , where \gamma =0.577\ldots is the Euler constant, and a few years later, Gronwall proved that \limsup \sigma (n)/(n\log \log n) is also equal to e^\gamma . Afterwards, several authors gave effective upper bounds for n/\varphi (n) and \sigma (n)/n, either under the Riemann hypothesis or without assuming it. Let X \ge 4 be a real number and \Phi (X) the maximum of n/\varphi (n) for n\le X. Similarly, we denote by \Sigma (X) the maximum of \sigma (n)/n for n\le X. Our first result gives effective upper and lower bounds for \Phi (X)/\Sigma (X). Next, we give new effective upper bounds for n/\varphi (n) and for \sigma (n)/n.