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In their book, ''Noncommutative Geometry, Quantum Fields and Motives,'' Alain Connes and Matilde Marcolli begin their preface by saying:

The unifying theme, which the reader will encounter in different guises throughout the book, is the interplay between noncommutative geometry and number theory, the latter especially in its manifestation through the theory of motives. For us, this interwoven texture of noncommutative spaces and motives will become a tool in the exploration of two spaces, whose role is central to many developments of modern mathematics and physics: ² Space-time and ² The set of prime numbers. One may be tempted to think that, looking from the vantage point of those who sit atop the vast edifice of our accumulated knowledge of such topics as space and numbers, we ought to know a great deal about these two spaces. However, there are two fundamental problems whose difficulty is a clear reminder of our limited knowledge, and whose solution would require a more sophisticated understanding than the one currently within our immediate grasp: ² The construction of a theory of quantum gravity (QG) and ² The Riemann hypothesis (RH). The purpose of this book is to explain the relevance of noncommutative geometry (NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall discover that there are deep analogies between these two problems which, if properly exploited, are likely to enhance our grasp of both of them.

Can someone explain in the simplest possible terms what really the link between the RH and QG that Connes and Marcolli were talking about ?

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    $\begingroup$ This pdf suggests it is about letting $q \to 1$ in the RH for curves over $\mathbb{F}_q$ to obtain it for "curves over $\mathbb{F}_1$". I don't understand what it means with the space of adeles classes of a global field being "non-commutative" $\endgroup$
    – reuns
    Commented Jan 26, 2019 at 11:17

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Not a direct answer, but maybe worth pointing out:

An apparently substantial insight into a relation between gravity and the zeros of the Riemann zeta-function (hence the Riemann hypothesis) was recently found via $p$-adic string theory by Shing-Tung Yau et al.:

  • An Huang, Bogdan Stoica, Shing-Tung Yau, "General relativity from $p$-adic strings" (arXiv:1901.02013)

Seems quite remarkable to me.

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