A CONDITIONAL DENSITY FOR CARMICHAEL NUMBERS

Abstract

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to$X$is$\gg X^{1-R}$, where$R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance’s conjectured density of$X^{1-R}$with$R=(1+o(1))\log \log \log X/\text{log}\log X$.