Small values of Carmichael's lambda function

Abstract

Let $λ(n)$ be the exponent of the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$, and set $L(x,y) = \#\{n\le x: λ(n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp((\log_2{x})^{1+ε}) \le y\le x/\exp((\log_2{x})^{1+ε})$. Our bound is asymptotically sharp under a plausible hypothesis on powersmooth shifted primes. As an application, we obtain a new upper bound on the count of odd $n\le x$ for which the order of $2$ modulo $n$ is appreciably smaller than $x^{1/2}$.