Selberg zeta function of a Riemann surface X is known to encode the discrete spectrum of the Laplacian on X via the Selberg trace formula. In this talk, following Lewis-Zagier, we will explain how one obtains the Selberg zeta function of the modular curve as the Fredholm determinant of an appropriate operator  on an appropriate Banach space. Along the way, we will discuss the close relationship between the operators in question and continued fractions. Should time permit, we will mention some ongoing work, partly joint with M. Fraczek, B. Mesland and M.H. Şengün.