Numbers of the form $kf(k)$

Abstract

For a function $f:\mathbb{N}\to \mathbb{N}$, 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 $\phi (n)=\#\{1\le k\le n:(k,n)=1\}$ be Euler's totient function. We prove that \begin{gather*} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! 1) \quad N^{\times}_τ(x) \asymp \frac{x}{(\log x)^{1/2}}; \\ 2) \quad N^{\times}_ω(x) = (1+o(1))\frac{x}{\log\log x}; \\ \!\!\!\!\!\!\!\!\! 3) \quad N^{\times}_φ(x) = (c_0+o(1))x^{1/2}, \end{gather*} where $c_0=\mathrm{1.365...}$\,.

arXiv commentThe error term in Theorem 1.2 is improved in this version of the paper