On the range of Carmichael's universal-exponent function

Florian Luca ¹   ; Carl Pomerance ²

¹ Mathematical Institute, UNAM Juriquilla Juriquilla, 76230 Santiago de Querétaro Querétaro de Arteaga, México and School of Mathematics University of the Witwatersrand P.O. Box Wits 2050 Johannesburg, South Africa
² Department of Mathematics Dartmouth College Hanover, NH 03755–3551, U.S.A.

Acta Arithmetica, Tome 162 (2014) no. 3, pp. 289-308

Résumé

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.

Détail

DOI : 10.4064/aa162-3-6

Keywords: lambda denote carmichaels function lambda universal exponent multiplicative group modulo closely related eulers varphi function here image lambda much denser image varphi particular number lambda values exceeds log large while varphi equal log old result erd improve earlier result first author friedlander giving upper bound distribution lambda values