Premium
On the Number of Carmichael Numbers up to x
Author(s) -
Harman Glyn
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004686
Subject(s) - mathematics , sieve (category theory) , mathematics subject classification , number theory , arithmetic , discrete mathematics , combinatorics
It is shown that, for all large x , there are more than x 0.33 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).