ncts

It is well-known that the Mellin transform of the (normalized) Jacobi theta function gives the completed Riemann zeta function. Seeking for an analogy of this phenomenon in the function field setting, recently, Green constructed a motivic map so that its particular values yield the special values of the Carlitz zeta function. In the present talk, we describe a generalization of this construction for Drinfeld modules of arbitrary rank. In particular, we introduce formulas for the logarithms. As a consequence, for a Drinfeld module defined over the finite base field, we relate the special values of the Goss L -function of to the image of this motivic map at the product of the vector of periods of with the rigid analytic trivialization of the corresponding dual t -motive. Our result can be seen as a characteristic
p -analogue of the integral representation of Hasse-Weil type zeta functions. This is a joint work with Nathan Green.