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Equidistribution rates, closed string amplitudes, and the Riemann hypothesis

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Abstract

We study asymptotic relations connecting unipotent averages of \( {\text{Sp}}\left( {2g,\mathbb{Z}} \right) \) automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.

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Authors and Affiliations

  1. Department of Physics and Mathematics, Università dell’Insubria, Via Valleggio 11, I-22100, Como, Italy

    Sergio L. Cacciatori

  2. INFN, Sezione di Milano, Via Celoria 16, I-20133, Milano, Italy

    Sergio L. Cacciatori

  3. Department of Physics, Università di Milano Bicocca, piazza della Scienza 3, I-20126, Milano, Italy

    Matteo Cardella

  4. Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9 A-1090, Vienna, Austria

    Matteo Cardella

Authors
  1. Sergio L. Cacciatori
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  2. Matteo Cardella
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Correspondence to Matteo Cardella.

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Cacciatori, S.L., Cardella, M. Equidistribution rates, closed string amplitudes, and the Riemann hypothesis. J. High Energ. Phys. 2010, 25 (2010). https://doi.org/10.1007/JHEP12(2010)025

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  • DOI: https://doi.org/10.1007/JHEP12(2010)025

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