Communications in Mathematical Physics

, Volume 277, Issue 1, pp 69–81

The Product Over All Primes is 4π2


DOI: 10.1007/s00220-007-0350-z

Cite this article as:
García, E.M. & Marco, R.P. Commun. Math. Phys. (2008) 277: 69. doi:10.1007/s00220-007-0350-z


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.

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute for the International Education of Students, IESMadridSpain
  2. 2.LAGA, CNRS UMR 7539Université Paris 13VilletaneuseFrance