| Abstract: For each prime number p, there exists a p-adic version of the Veneziano amplitude and its higher-point generalizations. In 1987, in an attempt to join together analytic number theory and string theory, Freund and Witten introduced adelic amplitudes obtained by multiplying together all $p$-adic amplitudes. In my talk I will provide an overview of this story, assuming no prior knowledge of p-adic numbers. Subsequently, I will demonstrate how for special kinematic configurations the adelic 5-point amplitude can be evaluated explicitly in terms of ratios of the Riemann zeta function, and how this result teaches us that adelic amplitudes are non-analytic. In light of this fact, it becomes necessary to reevaluate previous adelic product formulas in the physics literature, which rely on analytic continuation. I will propose an alternative formulation and discuss the unitarity and critical dimensions of the adelic amplitudes thereby obtained. |