ZETA-FUNCTIONS OF TWO-SIDED IDEALS IN ARITHMETIC ORDERS

The thesis deals with the theory of two-sided ideals in arithmetic orders. The theory and techniques developed by Bushnell and Reiner are used.;The work begins with an introduction to the theory of Z - and L-series. The basic plan is to compare these series with a Z-integral whose analytic properties are more accessible, and then use these properties to obtain some analogous ones of Z- and L-series. Next the theory of two-sided ideals is studied. First we translate the general theory just developed to our context; then we obtain explicit formulas for the zeta functions for some particular classes of orders, and we give some examples. We also study, in the simple case, the behavior of the zeta-functions at their largest pole. The thesis ends with discussion of some possible generalizations of the prime ideal theorem to two-sided ideals of arithmetic orders in simple algebras.