The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant function. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type L-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special derivative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.
We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules dened over the polynomial ring Fq[\theta]. This generalizes results of Anderson for the
rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.