Problem 1057

Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$ . Is it true that $C(x)=x^{1-o(1)}$ ?

Status

Open

No claim settles this problem.

Open. The site's label is OPEN, and no claim page is recorded.

AGP94: Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many Carmichael numbers. Ann. of Math. (2) (1994), 703-722.
Er56c: Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956), 201-206.
Gu04: Guy, Richard K., Unsolved problems in number theory, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. A13 "Carmichael numbers", printed p. 50: the report of Alford, Granville and Pomerance's infinitely many Carmichael numbers, more than $x^\beta$ below $x$ with $\beta>0.290306>2/7$ , and "Erdős had conjectured that $(\ln C(x))/\ln x$ tends to 1 as $x$ tends to infinity", with his upper bound improving Knödel; no proofs. Library home: Guy 2004.
Ha08: Harman, Glyn, Watt's mean value theorem and Carmichael numbers. Int. J. Number Theory (2008), 241-248.
Li22: J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
Po89: Pomerance, Carl, Two methods in elementary analytic number theory. In R. A. Mollin (ed.), Number Theory and Applications, Kluwer Academic Publishers (1989), 135-161.