Numbers of the form $kf(k)$

TL;DR

This paper investigates the distribution of numbers of the form n=kf(k) for specific arithmetic functions, revealing asymptotic behaviors for divisor, prime divisor, and Euler's totient functions.

Contribution

It provides new asymptotic formulas for the count of such numbers when f is the divisor, prime divisor, or totient function, advancing understanding of their distribution.

Findings

N^{ imes}_{ au}(x) hicksim x / (\log x)^{1/2}

N^{ imes}_{\omega}(x) hicksim x / \log \log x

N^{ imes}_{\varphi}(x) hicksim c_0 x^{1/2}

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 $c_0=1.365...$\,.