Communications in Mathematical Physics

, Volume 277, Issue 1, pp 69-81

First online:

The Product Over All Primes is 4π2

  • E. Muñoz GarcíaAffiliated withInstitute for the International Education of Students, IES
  • , R. Pérez MarcoAffiliated withLAGA, CNRS UMR 7539, Université Paris 13 Email author 

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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.