On the range of Carmichael's universal-exponent function
Volume 162 / 2014
Acta Arithmetica 162 (2014), 289-308
MSC: Primary 11N37.
DOI: 10.4064/aa162-3-6
Abstract
Let $\lambda $ denote Carmichael's function, so $\lambda (n)$ is the universal exponent for the multiplicative group modulo $n$. It is closely related to Euler's $\varphi $-function, but we show here that the image of $\lambda $ is much denser than the image of $\varphi $. In particular the number of $\lambda $-values to $x$ exceeds $x/(\log x)^{.36}$ for all large $x$, while for $\varphi $ it is equal to $x/(\log x)^{1+o(1)}$, an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of $\lambda $-values.
