On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms

Details On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms

2007-11-04

arxiv.org/abs/0711.0499v1

Mathematics

Number Theory

14pages

Scientific paper

We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship.

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