Abstract
We generalize the classical definition of zeta-regularization of an infinite product. The extension enjoys the same properties as the classical definition, and yields new infinite products. With this generalization we compute the product over all prime numbers answering a question of Ch. Soulé. The result is 4π2. This gives a new analytic proof, companion to Euler’s classical proof, that the set of prime numbers is infinite.
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References
Aigner, M., Ziegler, G.M.:Proofs from the book. Berlin-Heidelberg, New York: Springer Verlag, 2nd Edition, 2000
Borel, É.: Leçons sur les series divergentes. Paris: Gauthier-Villars, 1928
Dahlquist G. (1951). On the analytic continuation of Eulerian products. Arkiv för Matematik 1: 533–554
Deninger, C.: Some analogies between Number Theory and Dynamical Systems on foliated spaces. Doc. Mat. J. DMV Extra Volume ICM I, 163–186 (1998)
Elizalde, E., Odintsov, S.D., Romeo, A., Bytsenko, A.A., Zerbini, S.: Zeta regularization techniques with applications. Singapore, World Scientific, 1994
Euclid L. (1956). The Thirteen Books of the Elements, Translated with introduction and commentary by Thomas L. Heath, Vol. 1, 2, 3. Dover Publications, New York
Euler, L.: Introductio in Analysin Infinitorum. Facsimil edition and commented translation of the edition of 1748 stored in the library of the Real Instituto y Observatorio de la Armada en San Fernando, eds. A.J. Duran Guardeño, F.J. Pérez Fernández Bacelona: Real Socieded Mathematic Espazola, 2000
Hardy G.H. (1963). Divergent series. Clarendon Press, Oxford
Illies G. (2001). Regularized products and determinants. Commun. Math. Phys. 220: 69–94
Koblitz, N.: p-adic numbers, p-adic analysis and zeta functions. Graduate Texts in Mathematics 58, 2nd edition, Berlin Heidelberg-New York: Springer, 1998
Landau E. and Walfisz A. (1920). Über Die Nichtfortsetzbarkeit Einiger Durch Dirichletsche Reihen Definierter Funktionen. Rendiconti del Circolo Matematico di Palermo 44: 82–86
Muñoz Garcia, E., Pérez Marco, R.: The product over all prime numbers is 4π 2. Preprint IHES M/03/34, www.ihes.fr, May 2003
Muñoz Garcia, E., Pérez Marco, R.: Super-regularization of infinite products. Preprint IHES M/03/52, www.ihes.fr, August 2003
Ramis, J.-P.: Séries divergentes et théories asymptotiques. In: Panoramas et synthèses, Paris: Société Mathématique de France, 1993
Remmert, R.: Classical topics in Complex Function Theory. Graduate Texts in Mathematics 172, Berlin, Heidelberg-New York: Springer Verlag
Ray D. and Singer I. (1973). Analytic torsion for analytic manifolds. Ann. Math. 98: 154–177
Soulé C., Abramovich D., Burnol J.-F. and Kramer J. (1992). Lectures on Arakelov Geometry. Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge
Siu Y.-T. (1974). Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York
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Communicated by A. Connes
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García, E.M., Marco, R.P. The Product Over All Primes is 4π2 . Commun. Math. Phys. 277, 69–81 (2008). https://doi.org/10.1007/s00220-007-0350-z
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DOI: https://doi.org/10.1007/s00220-007-0350-z