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MCA where M ranges over all full left ideals of A. The series converges for Re(s) > 1. Here, (A : M) is the number of elements in A/M, and s is a complex variable. For the case where A = R, the above is the usual Dedekind zeta function of R, namely, where a ranges over all nonzero ideals of R. Let P range over all maximal ideals of R, and let Rp, Apy etc., denote P-adic completions. Solomon showed easily that fA(») = n rApto. p and introduced a "global" zeta function fA(s), which depends on A and R but not on A, with the property that the P-part of ÇA(s) coincides with ?AJ>(S) f°r almost all P. (To be explicit, this occurs whenever Ap is a direct sum of full matrix algebras over fields, and Ap is a maximal .Rp-order in Ap.) Solomon's conjectures involve the comparison between ÇA(s) and ÇA(s) at arbitrary P's. Let us place the above in the more general setting used by Solomon. Let L be a full A-lattice in an A -module V, and define

Solomon’s conjectures and the local functional equation for zeta functions of orders