We consider maximally chaotic dynamical system defined on the fundamental domain of the modular group.This fundamental domain on the Poincare half plane has finite volume and infinite extension in the vertical axis that correspond to a cusp.In classical regime the geodesic flow in this domain represents one of the most chaotic dynamical systems, has mixing of all orders, Lebesgue spectrum and non-zero Kolmogorov entropy. The classical correlation functions decay exponentially with an exponent proportional to the entropy. Here we calculated quantum mechanical two- andfour-point correlation functions. The two-point correlation function decays exponentially and the out-of-timefour-point correlation function decay to almost zero value and then increases with the subsequent large fluctuations. This confirms the existence of the scrambling time in this maximally chaotic system. We demonstrate that the zeros of the Riemann zeta function define the positions and the widths of the resonances of the quantum mechanical system. e-Print: arXiv:1802.04543 e-Print: arXiv:1808.02132 e-Print: arXiv:1809.09491