Divisor function for quaternion algebras and application to fourth moments of L-functions

In this paper, we define the divisor function for the quaternion algebra over Q which ramifies precisely at p and ∞. For the zeta function of a maximal order, we prove a quaternion analogue of the well-known formula . As an application, we obtain an average of fourth moments of L-functions of newforms with respect to Γ0(p) with the trivial character, following Duke's method. Due to the fact that the class number is no longer one, we need to consider a system of Dirichlet series and a system of automorphic functions in a hyperbolic (n+1)-space of signature (n,1).