On a sum involving certain arithmetic functions and the integral part function (Q2696318)

Let \(\sigma\) be the sum-of-divisors function, \(\beta\) be the alternating sum-of-divisors function, \(\varphi\) be the Euler totient function, and \(\Psi\) be the Dedekind function. In the paper under review, the authors prove that if \(f\in\{\sigma, \beta, \varphi, \Psi\}\), then as \(x\to\infty\), \[ \sum_{n\leq x}\frac{f([x/n])}{[x/n]}=C_f\,x+O(x^{1/4+\varepsilon}), \] where \(C_f=\sum_{n\geq 1}f(n)/(n^2(n+1))\) and \(\varepsilon>0\).