  We prove the existence of secondary terms of order X^{5/6} in the
Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of
quadratic fields. For cubic fields this confirms a conjecture of
Datskovsky-Wright and Roberts. We also prove a variety of generalizations,
including to arithmetic progressions, where we discover a curious bias in the
secondary term. Roberts' conjecture has also been proved independently by
Bhargava, Shankar, and Tsimerman. In contrast to their work, our proof uses the
analytic theory of zeta functions associated to the space of binary cubic
forms, developed by Shintani and Datskovsky-Wright.
