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

Secondary terms in counting functions for cubic fields