Abstract
The present paper is concerned with some equalities between zeta functions of quaternion algebras introduced in Godement [6], Shimura [11], Tamagawa [13]. Let A be a quaternion algebra over a totally real algebraic number field 1? of degree m, and let D be an order in A; let S be the idele group of A, and U the group of units in S with respect to D; let p be a representation of ( a/f)*, f being an integral two-sided rD-ideal, and let ki (1 < i < m) be non-negative integers. These being given, we can speak of a space of automorphic functions associated with (p, {ki}) (cf. ? 2.2 and ? 2.5) and of a repre-sentation Z of the Hecke ring 9R(U, @) in this space. Let T(q) be the sum of all integral elements in 9R(U, @ of norm q and let C(s) be the Dirichlet. series defined by
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