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The Product Over All Primes is 4π2

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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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Authors and Affiliations

  1. Institute for the International Education of Students, IES, Avda Seneca 7, 28040, Madrid, Spain

    E. Muñoz García

  2. LAGA, CNRS UMR 7539, Université Paris 13, 93430, Villetaneuse, France

    R. Pérez Marco

Authors
  1. E. Muñoz García
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  2. R. Pérez Marco
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Correspondence to R. Pérez Marco.

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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

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