Popular subsets for Euler’s $$\varphi $$-function

Abstract

Let $$\varphi (n) = \#(\mathbb {Z}/n\mathbb {Z})^{\times }$$ (Euler’s totient function). Let $$\epsilon > 0$$ , and let $$\alpha \in (0,1)$$ . We prove that for all $$x > x_0(\epsilon ,\alpha )$$ and every subset $$\mathscr {S}$$ of [1, x] with $$\#\mathscr {S}\le x^{1-\alpha }$$ , the number of $$n\le x$$ with $$\varphi (n)\in \mathscr {S}$$ is at most $$x/L(x)^{\alpha -\epsilon }$$ , where $$\begin{aligned} L(x) = \exp (\log x\cdot \log _3{x}/\log _2 x). \end{aligned}$$ Under plausible conjectures on the distribution of smooth shifted primes, this upper bound is best possible, in the sense that the number $$\alpha $$ appearing in the exponent of L(x) cannot be replaced by anything larger.