The aim of this book is to provide an introduction to quadratic forms that builds from basics up to the most recent results. Professor Kitaoka is well known for his work in this area, and in this book he covers many aspects of the subject, including lattice theory, Siegel's formula, and some results involving tensor products of positive definite quadratic forms. The reader is required to have only a knowledge of algebraic number fields, making this book ideal for graduate students and researchers wishing for an insight into quadratic forms.

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We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

YOSHIYUKI KITAOKAOur purpose is to prove the following THEOREM. Let k be an even integer > 6. Let f{Z) = Σ αCF)e(tr TZ) be a Siegel cusp form of degree two, weight k. Then we haveThis was announced in [3] where we put an assumption on estimates of generalized Kloosterman sums. Here, we give a complete proof with a proof of that assumption.Every cusp form of degree two, weight k > 6 (k = 0 mod 2) is a linear combination of Poincare series [1,4]. Using their rather formal Fourier expansion given in [1], we prove our theorem.Notation. By Z, Q, R and C we denote the ring of rational integers, the field of rational numbers, the field of real numbers, and the field of complex numbers, repsectively. H denotes the upper half-plane of genus two:

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