The Hattori-Stallings rank, the Euler-Poincar\'e characteristic and zeta functions of totally disconnected locally compact groups

For a unimodular totally disconnected locally compact group G we introduce and study an analogue of the Hattori-Stallings rank (P) G for a finitely generated projective rational discrete left QG-module P. Here hG denotes the Q-vector space of left invariant Haar measures of G. Indeed, an analogue of Kaplansky's theorem holds in this context (cf. Theorem A). As in the discrete case, using this rank function it is possible to define a rational discrete Euler-Poincar\'e characteristic G whenever G is a unimodular totally disconnected locally compact group of type FP_ of finite rational discrete cohomological dimension. E. g. , when G is a discrete group of type FP, then G coincides with the ''classical'' Euler-Poincar\'e characteristic times the counting measure \₁\. For a profinite group O, ₎ equals the probability Haar measure ₎ on O. Many more examples are calculated explicitly (cf. Example 1. 7 and Section 5). In the last section, for a totally disconnected locally compact group G satisfying an additional finiteness condition, we introduce and study a formal Dirichlet series _₆, ₎ (s) for any compact open subgroup O. In several cases it happens that _₆, ₎ (s) defines a meromorphic function _₆, ₎ C C of the complex plane satisfying miraculously the identity G=_₆, ₎ (-1) ^-1₎. Here ₎ denotes the Haar measure of G satisfying ₎ (O) =1.