Abstract: Many of the deepest and most far reaching results in number theory come from blending together algebraic and analytic techniques, for example, expressing a meaningful arithmetic quantity as an integral. This is called an integral representation. But what should you do when the analytic side is missing? In my talk, I will sketch a couple areas in classical number theory where this analytic theory is well developed, such as values of the Riemann zeta function and multiple zeta values, and explain what we learn from the analytic side. Then, I will describe a setting - characteristic p valued function fields and Drinfeld modules - where this integration theory is missing. Finally, I will describe my contributions towards developing an algebraic replacement for this missing integration theory in characteristic p and discuss what these results tell us.