On the Error Terms of Chebyshev Functions for SL4

Abstract

Our object of research are certain higher order counting functions of Chebyshev type, associated to the compact symmetric space SL4. In particular, we consider the functionψ1(x)resp.ψ3(x), of order1resp.3.As it is well known, any such function can be represented as a sum of some explicit part, and the corresponding error term. The explicit part is usually indexed over singularities of the attached Selberg zeta functions, while the error term depends on the dimension of the underlying symmetric space. Thus, these functions generalize the classical yes functionπ(x)counting prime geodesics of appropriate length. More precisely, the Chebyshev functions divided by adequate power of x, represent quite natural approximations for the functionπ(x). In this research, we are particularly interested in the error terms ofψ1(x)/xandψ3(x)/x3.