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We determine the mean number of 3-torsion elements in the class groups ofquadratic orders, where the quadratic orders are ordered by their absolutediscriminants. Moreover, for a quadratic order $\mathcal{O}$ we distinguishbetween the two groups: $\mathrm{Cl}_3(\mathcal{O})$, the group of idealclasses of order $3$; and $\mathcal{I}_3(\mathcal{O})$, the group of ideals oforder $3$. We determine the mean values of both $|\mathrm{Cl}_3(\mathcal{O})|$and $|\mathcal{I}_3(\mathcal{O})|$, as $\mathcal{O}$ ranges over any family oforders defined by finitely many (or in suitable cases, even infinitely many)local conditions.  As a consequence, we prove the surprising fact that the mean value of thedifference $|\mathrm{Cl}_3(\mathcal{O})|-|\mathcal{I}_3(\mathcal{O})|$ is equalto $1$, regardless of whether one averages over the maximal orders in complexquadratic fields or over all orders in such fields or, indeed, over any familyof complex quadratic orders defined by local conditions. For any family of realquadratic orders defined by local conditions, we prove similarly that the meanvalue of the difference$|\mathrm{Cl}_3(\mathcal{O})|-\frac13|\mathcal{I}_3(\mathcal{O})|$ is alwaysequal to $1$, independent of the family.

The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders