Class numbers of orders in cubic fields

In this paper it is shown that the sum of class numbers of orders to complex cubic fields obeys an asymptotic law similar to the prime numbers as the bound on the regulators tends to infinity. Here only orders are considered which are maximal at two given primes This result extends work of Sarnak in the real quadratic case. It seems to be the first asymptotic result on class numbers for number fields of degree higher than two. (C) 2002 Elsevier Science (USA).