Numbers of the form $kf(k)$ (2201.09287v2)

Abstract: For a function $f\colon \mathbb{N}\to\mathbb{N}$, define $N^{\times}_{f}(x)=#{n\leq x: n=kf(k) \mbox{ for some $k$} }$. Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and $\varphi(n)=#{1\leq k\leq n: (k,n)=1 }$ be Euler's totient function. We prove that \begin{gather*} !!!!!!!!!!!!!!!!!!!!!!! 1) \quad N^{\times}_{\tau}(x) \asymp \frac{x}{(\log x)^{1/2}}; \ 2) \quad N^{\times}_{\omega}(x) = (1+o(1))\frac{x}{\log\log x}; \ !!!!!!!!! 3) \quad N^{\times}_{\varphi}(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=1.365...$\,.