Small values of Carmichael's lambda function

Paul Pollack

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arXiv · 2606.19132

Small values of Carmichael's lambda function

Paul Pollack

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Abstract

Let $\lambda(n)$ be the exponent of the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$, and set $L(x,y) = \#\{n\le x: \lambda(n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp((\log_2{x})^{1+\epsilon}) \le y\le x/\exp((\log_2{x})^{1+\epsilon})$. 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 $x^{1/2}$.

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Paul Pollack. 2026-06-17. Small values of Carmichael's lambda function. https://arxiv.org/abs/2606.19132

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