Two Contradictory Conjectures Concerning Carmichael Numbers

Erdos conjectured that there are alpha sup (1-0(1)) Carmichael numbers up to alpha, whereas Shanks was skeptical as to whether one might even find an alpha up to which there are more than sqrt alpha Carmichael numbers. Alford, Granville and Pomerance showed that there are more than alpha sup (2/7) Carmichael numbers up to alpha, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdos must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data and so we herein derive conjectures that are consistent with Shanks's observations, while fitting in with the viewpoint of Erdos and the results of [2,3].