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On the range of Carmichael's universal-exponent function

Volume 162 / 2014

Florian Luca, Carl Pomerance Acta Arithmetica 162 (2014), 289-308 MSC: Primary 11N37. DOI: 10.4064/aa162-3-6


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.


  • Florian LucaMathematical Institute, UNAM Juriquilla
    Juriquilla, 76230 Santiago de Querétaro
    Querétaro de Arteaga, México
    School of Mathematics
    University of the Witwatersrand
    P.O. Box Wits 2050
    Johannesburg, South Africa
  • Carl PomeranceDepartment of Mathematics
    Dartmouth College
    Hanover, NH 03755–3551, U.S.A.

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