Regularized adelic formulas for string and superstring amplitudes in one-class quadratic fields

We obtain regularized adelic formulas for gamma and beta functions for fields of rational numbers and the one-class quadratic fields and arbitrary quasicharacters (ramified or not). We consider applications to four-tachyon tree string amplitudes, generalized Veneziano amplitudes (open string), perturbed Virasoro amplitudes (closed string), massless four-particle tree open and closed superstring amplitudes, Ramond–Neveu–Schwarz superstring amplitudes, and charged heterotic superstring amplitudes. We establish certain relations between different string and superstring amplitudes.