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A CONDITIONAL DENSITY FOR CARMICHAEL NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 13 February 2020, pp. 379-388
- Print publication:
- June 2020
-
- Article
- Export citation
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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 
$R=(1+o(1))\log \log \log X/\text{log}\log X$.
