There is a presumably very deep conjecture that there are infinitely many primes for which the quadratic field has class number 1. The authors’ principal result gives a comparatively modest but unconditional approach to this conjecture, by showing that there are infinitely many such for which the ideal class group of has no element of order 3. Their result is quantitative in that it shows that a positive proportion of these primes not exceeding possesses the stated property.
Let be a fundamental discriminant, so that it is squarefree and is either or is , where . Let denote the number of negative with , and define similarly. Let denote the number of th roots of unity in , so that , where is the -rank of the title. Let . The authors establish an asymptotic expression for when and the sum is restricted to in one of . The exponent is better than that appearing in the earlier paper of K. Belabas [Ann. Inst. Fourier 46, No. 4, 909–949 (1996; Zbl 0853.11088)].
As in the earlier paper, this information can be used as the input for a sifting argument. The authors now infer, for example, that there are infinitely many fundamental discriminants for which the -ranks satisfy and . By means of a sieve with weights the constant 6 can be replaced by 3.
(This is the same review as for Math. Rev.).