Let d be a square-free number and let CL(−d) denote the ideal class group of the
imaginary quadratic number field ℚ(√−d). Further let
h(−d) = #CL(−d) denote
the class number. For integers g [ges ] 2, we define [Nscr ]g(X)
to be the number of square-free d [ges ] X such that CL(−d) contains
an element of order g. Gauss' genus theory
demonstrates that if d has at least two odd prime factors (in particular, for almost all
d) then CL(−d) contains ℤ2 as a subgroup. Thus
N2(X) ∼ 6X/π2. The behaviour of
[Nscr ]g(X) is not understood for any other value of g.
It is believed that [Nscr ]g(X) ∼ CgX
for some positive constant Cg. For odd primes g, H. Cohen
and H. Lenstra  conjectured that
N. Ankeny and S. Chowla  first showed that
[Nscr ]g(X) → ∞ as X → ∞. Although
they did not point this out, their method demonstrates that
[Nscr ]g(X) [Gt ] X1/2. Recently, M. R. Murty 
improved this to [Nscr ]g(X) [Gt ] X1/2+1/g.
Hitherto this represented the best known lower bounds for [Nscr ]g(X) except in
the cases g = 4 and g = 8. In the cases g = 4 or 8, P. Morton 
used class field theory techniques to show that
[Nscr ]g(X) [Gt ] X1−ε.
In fact, he demonstrated the elegant result that given any non-negative integers r, s
and t, there are ‘many’ d with
ℤr2 × ℤs4 × ℤt8
(see  for a precise statement). The complementary question of finding d with
p [nmid ] h(−d) has also
attracted a lot of attention. H. Davenport and H. Heilbronn  proved the striking
result that the proportion of d with 3 [nmid ] h(−d)
is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono 
have shown that there are [Gt ] √X/log X square-
free integers d [les ] X such that p [nmid ] h(−d).
In this paper, we sharpen Murty's lower bounds on [Nscr ]g(X)
for all values of g; see Theorem 1 below. We also offer a simple proof that
[Nscr ]4(X) [Gt ] X/√log X; see
Proposition 2 below. In §5 we express the hope that these methods may lead to
[Nscr ]3(X) [Gt ] X1−ε.