Geodesic
Adelic formulae for the gamma and beta functions of completions of algebraic number fields, and applications of them to string amplitudes
Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 67-90

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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. 
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     author = {V. S. Vladimirov},
     title = {Adelic formulae for the gamma and beta functions of completions of algebraic number fields, and applications of them to string amplitudes},
     journal = {Izvestiya. Mathematics },
     pages = {67--90},
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     volume = {60},
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     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a2/}
}
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V. S. Vladimirov. Adelic formulae for the gamma and beta functions of completions of algebraic number fields, and applications of them to string amplitudes. Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 67-90. http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a2/