The author makes a substantial contribution to the investigation of a general ''zeta function'' introduced by \textit{L. Solomon} [Adv. Math. 26, 306-326 (1977; Zbl 0374.20007)]. In particular, the definition of Solomon's zeta function replaces the ring R of all algebraic integers in an algebraic number field K by an R-order \(\Lambda\) in a finite- dimensional semisimple K-algebra A. The problems considered previously involve mainly the one-sided ideal theory, of \(\Lambda\), and the methods of investigation involve representation theory, harmonic analysis on locally compact groups, and algebraic number theory along the lines of \textit{J. Tate}'s thesis [see Chapter XV of ''Algebraic number theory'', ed. by J. W. S. Cassels and A. Fröhlich (1967; Zbl 0153.074)]. Apart from the initial contribution of L. Solomon (loc. cit.), the methods and results of the theory were almost entirely due to \textit{C. J. Bushnell} and \textit{I. Reiner} [Math. Z. 173, 135-161 (1980; Zbl 0438.12004); J. Reine Angew. Math. 327, 156-183 (1981; Zbl 0455.12010) and ibid. 329, 88-124 (1981; Zbl 0488.12011)], and other papers which are too numerous to list here in detail. The main new contribution of the present article is to the theory in the case of two-sided ideals in arithmetic orders. Amongst many results of a similar type, two major ones are as follows: i) Let \(\Lambda\) be a \({\mathbb{Z}}\)-order in a simple \({\mathbb{Q}}\)-algebra A, let I be a two-sided ideal in the two-sided genus \(g^ e(\Lambda)\) of \(\Lambda\) and let J range over all maximal two-sided ideals of \(\Lambda\) which are isomorphic to I. Then \[ \sum_{J}(\Lambda:J)^{-s}=h^{- 1}_{\Lambda}\quad \log \{1/(s-n^{-2})\}+G_ 0(s), \] where \(n^ 2=\dim A\), \(G_ 0(s)\) is holomorphic for \(Re(s)\geq n^{-2}\), and \(h_{\Lambda}\) is the number of isomorphic classes in \(g^ e(\Lambda).\) (ii) The number of ideals J as above such that \((\Lambda:J)