Small values of Carmichael's lambda function

Abstract

Let $\hat{I}»(n)$ be the exponent of the multiplicative group $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, and set $L(x,y)=\#\{n\le x:\hat{I}»(n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp (({\log }_{2}x{)}^{1+\hat{I}\mu })\le y\le x/\exp (({\log }_{2}x{)}^{1+\hat{I}\mu })$. 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 $x^{1/2}$.

20 pages