On common values of $\phi(n)$ and $\sigma(m)$, II

Abstract

For each positive-integer valued arithmetic function $f$, let ${\mathcal{V}}_f\subset \mathbb{N}$ denote the image of $f$, and put ${\mathcal{V}}_f\left(x\right):={\mathcal{V}}_f\cap \left[1,x\right]$ and ${\mathcal{V}}_f\left(x\right):=\#{\mathcal{V}}_f\left(x\right)$. Recently Ford, Luca, and Pomerance showed that ${\mathcal{V}}_{\varphi }\cap {\mathcal{V}}_{\sigma }$ is infinite, where $\varphi$ denotes Euler’s totient function and $\sigma$ is the usual sum-of-divisors function. Work of Ford shows that $V_{\varphi }\left(x\right)\asymp V_{\sigma }\left(x\right)$ as $x\to \infty$. Here we prove a result complementary to that of Ford et al. by showing that most $\varphi$-values are not $\sigma$-values, and vice versa. More precisely, we prove that, as $x\to \infty$,

$$\#\left\{n\le x:n\in {\mathcal{V}}_{\varphi }\cap {\mathcal{V}}_{\sigma }\right\}\le \frac{V_{\varphi }\left(x\right)+V_{\sigma }\left(x\right)}{{\left(loglogx\right)}^{1∕2+o\left(1\right)}}.$$