Derivation of Freund - Witten adelic formula for four-point Veneziano amplitudes

Abstract

On the basis of an analysis on the adelic group (Tate's formula) a regularization is proposed for the divergent infinite product ofp-adic Γ functions:

$$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{1 - p^{ - \alpha } }}, p = 2,3,5...,$$

and the adelic formula

$$reg\prod\limits_{p = 2}^\infty {\Gamma p(\alpha ) = \frac{{(\zeta \alpha )}}{{\zeta (1 - \alpha )}},} $$

where ζ(α) is the Riemann ζ function, is proved.