Adelic formulae for the gamma and beta functions of completions of algebraic number fields, and applications of them to string amplitudes

On the basis of analysis on the adele group (the Tate formula) of an algebraic number field, a regularization is constructed for the divergent adelic products of the gamma and beta functions of all (non-isomorphic) completions of this field. The formulae obtained are applied to representations of the four-point crossing-symmetric Veneziano amplitudes and Virasoro–Shapiro amplitudes in terms of regularized adelic products of string amplitudes (for open or closed strings) corresponding to all non-Archimedean completions of the algebraic number field under consideration.