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 Σ∞n=1 d(n)2n-s = ζ(s)4/ζ(2s). 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). © 2009 Elsevier Inc. All rights reserved.
Kim, H. H., & Zhang, Y. (2009). Divisor function for quaternion algebras and application to fourth moments of L-functions. Journal of Number Theory, 129(12), 3000–3019. https://doi.org/10.1016/j.jnt.2009.05.009