A CONDITIONAL DENSITY FOR CARMICHAEL NUMBERS.

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 $R=(1+o(1))\log \log \log X/\text{log}\log X$. [ABSTRACT FROM AUTHOR]