  In this article, we establish in a rigorous mathematical way that
Koba-Nielsen amplitudes defined on any local field of characteristic zero are
bona fide integrals that admit meromorphic continuations in the kinematic
parameters. Our approach allows us to study in a uniform way open and closed
Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In
the regularization process we use techniques of local zeta functions and
embedded resolution of singularities. As an application we present the
regularization of p-adic open string amplitudes with Chan-Paton factors and
constant B-field. Finally, all the local zeta functions studied here are
partition functions of certain 1D log-Coulomb gases, which shows an interesting
connection between Koba-Nielsen amplitudes and statistical mechanics.
