On the Number of Carmichael Numbers up to x

It is shown that, for all large x, there are more than x0.33 Carmichael numbers up to x, improving on the ground-breaking work of Alford, Granville and Pomerance, who were the first to demonstrate that there are infinitely many such numbers. The same basic construction as that employed by these authors is used, but a slight modification enables a stronger result on primes in arithmetic progressions based on a sieve method to be employed. 2000 Mathematics Subject Classification 11N13 (primary), 11N36 (secondary).