The precise form for the remainder term for is due to Shintani ([10]), improving on Davenport's original result. A remainder term for was proved by the first author in [1], and the present one will appear in [2].

We could still gain a little by avoiding the empty loops that we have mentioned. For this, we would need to find the exact range of values of a, then of b given a, and finally of c, given a and b. The result involves a very large number of cases and algebraic equations of large degree, and the gain is marginal, hence we do not do it. We can also easily characterize subclasses of real cubic fields. For example, we have the following:

Proposition 5.7

Let K be a totally real cubic number field, the unique reduced form associated to K and its Hessian. Then:

• K is cyclic ( i.e ) if and only if .
• if and only if or .
• if and only if or .
• if and only if , or .