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New asymptotic formulas for the distribution of left ideals of orders.

where X ranges over all left ideals of Λ which are isomorphic to M s Λ-modules. Landau's theorem gives an asymptotic formula of the required type for the partial sums of the coefficients of a (generalised) Dirichlet series, provided that the series satisfies a certain long list of hypotheses. The main group of these hypotheses demands that the series be continuable to a meromorphic function on the whole s-plane, and that it satisfy a certain functional equation. Such results for the series ΖΛ(Μ, s) are available from [6].