Diophantine approximation with multiplicative functions (Q967595)

Let \(\sigma(n)\) and \(\varphi(n)\) denote the sum of divisors function and Euler's totient function, respectively. The main result of this paper is that, given any real number \(\alpha>1\) and \(\varepsilon>0\), there are infinitely many positive integers \(n\) with \[ \left|\frac{\sigma(n)}{n}-\alpha\right|<n^{\varepsilon+\rho-1}, \] where \(\rho=(25-\sqrt{305})/16=0.470984\dots\). Moreover, if the Riemann hypothesis holds then \(\rho-1\) can be replaced with \(\tau-1\), where \(\tau=(3-\sqrt{5})/2=0.38196\dots\). In addition, \(\sigma(n)/n\) can be replaced with \(n/\phi(n)\), and the result also holds for \(0<\alpha<1\) with approximating functions \(n/\sigma(n)\) or \(\varphi(n)/n\). The author proves the result by modifying an argument given by \textit{D. Wolke} [Monatsh. Math. 83, 163--166 (1977; Zbl 0353.10004)] which in its original form could not lead to an exponent greater than \(1/2\). The basic new idea is to use the fact that there cannot be very many long intervals not containing a proposition of the correct number of primes.