On the range of Carmichael's universal-exponent function

Florian Luca, Carl Pomerance
2014 Acta Arithmetica  
Let λ denote Carmichael's function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler's ϕ-function, but we show here that the image of λ is much denser than the image of ϕ. In particular the number of λ-values to x exceeds x/(log x) .36 for all large x, while for ϕ 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 1 Mathematics Subject Classification: 11N37
doi:10.4064/aa162-3-6 fatcat:damb2ettqzfbhomhlvayvsvej4