Did Andre Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis? If so, what are a reference and/or a quotation?

  • $\begingroup$ If he did, I'd be surprised if he used those words, since they postulate a false dichotomy. "Prime number theory" and "analysis" are surely not disjoint? $\endgroup$ – Yemon Choi Dec 31 '10 at 0:48
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    $\begingroup$ For example, Li's analytic criterion for the RH does not involve prime numbers. Does that help you to understand my question a little better, Choi? $\endgroup$ – Jonathan Sondow Dec 31 '10 at 3:36
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    $\begingroup$ Thanks, that does indeed clarify, Sondow (although I still feel that if Weil made such a pronouncement, rhetoric was in danger of overwhelming the mathematical point being made). $\endgroup$ – Yemon Choi Dec 31 '10 at 5:06

The passage that comes to mind is from Weil's essay "L'avenir des mathematiques," which is in the first volume of his collected works.

“L’hypothèse de Riemann, après qu’on eut perdu l’espoir de la démontrer par les méthodes de la théorie des fonctions, nous apparaît aujourd’hui sous un jour nouveau, qui la montre inséparable de la conjecture d’Artin sur les fonctions L, ces deux problèmes étant deux aspects d’une même question arithmético-algébrique, où l’étude simultanée de toutes les extensions cyclotomiques d’un corps de nombres donné jouera sans doute le rôle décisif. L’arithmétique gaussienne gravitait autour de la loi de réciprocité quadratique; nous savons maintenant que celle-ci n’est qu’un premier example, ou pour mieux dire le paradigme, des lois dites “du corps de classe”, qui gouvernent les extensions abéliennes des corps de nobres algébriques; nous savons formuler ces lois de manière à leur donner l’aspect d’un ensemble cohérent; mais, si plaisante à l’œil que soit cette façade, nous ne savons si elle ne masque pas des symmétries plus cachées. Les automorphismes induits sur les groupes de classes par les automorphismes du corps, les propriétés des restes de normes dans les cas non cycliques, le passage à la limite (inductive ou projective) quand on remplace le corps de base par des extensions, par example cyclotomiques, de degré indéfiniment croissant, sont autant de questions sur lesquelles notre ignorance est à peu près complète, et dont l’étude contient peut-être la clef de l’hypothese de Riemann; étroitement liée à celles-ci est l’étude du conducteur d’Artin, et en particulier, dans le cas local, la recherche de la représentation dont la trace s’exprime au moyen des caractères simples avec des coefficients égaux aux exposants de leurs conducteurs. Ce sont là quelques-unes des directions qu’on peut et qu’on doit songer à suivre afin de pénétrer dans le mystère des extensions non abéliennes; il n’est pas impossible que nous touchions là à des principes d’une fécondité extraordinaire, et que le premier pas décisif une fois fait dans cette voie doive nous ouvrir l’accès à de vastes domaines dont nous soupçonnons à peine l’existence; car jusqu’ici, pour amples que soient nos généralisations des résultats de Gauss, on ne peut dire que nous les ayons vraiment dépassés.”

Edit: I found an official English translation.

"The Riemann hypothesis, after the attempts to prove it by function-theoretic methods had been given up, appears to-day in a new light, which shows it to be closely connected with the conjecture of Artin on the L-functions, thus making these two problems two aspects of the same arithmetico-algebraic question, in which the simultaneous study of all the cyclotomic extensions of a given number field will undoubtedly play a decisive role. Gaussian arithmetic was centered around the law of quadratic reciprocity; we know now that this law is only a first example, we might better say the pattern, the laws of "class fields," which control the abelian extensions of algebraic number-fields; we know how to formulate these laws so as to make them look like a coherent set. But, pleasant as this facade may be to the eye, we do now know whether it might not hide deeper lying symmetries. The automorphisms induced in the class groups by the automorphisms of the field, the properties of the norm-residues in the non-cyclic cases, the passage to the limit (inductive or projective) when the base field is replaced by extensions, for example, cyclotomic extensions, of indefinitely increasing degree, all these are questions on which our ignorance is almost complete and in whose study the key to the Riemann hypothesis is perhaps to be found. Closely connected with these questions is the study of Artin's conductor and, in particular, in the local case, the search for the representation, whose trace can be expressed by means of simple characters with coefficients equal to the exponents of their conductors. These are some of the directions which can and must be followed up in order to penetrate the mystery of non-abelian extensions; it is not impossible that we are here close to principles of extraordinary fertility and that, once the first decisive step on this road will have been taken, we shall gain access to vast domains whose existence is hardly suspected. For, however wide our generalizations of Gauss' results may be, we can hardly claim to have as yet really moved beyond them."

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    $\begingroup$ Some of us had read this passage before blogs were born. $$ $$ $\endgroup$ – Chandan Singh Dalawat Dec 31 '10 at 9:06
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    $\begingroup$ I agree that Le Bruyn's blog post appears to be the immediate source of this quote: they both have exactly the same two spelling mistakes. $\endgroup$ – Chandan Singh Dalawat Dec 31 '10 at 9:26
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    $\begingroup$ Dear Ed, I'm very happy that you refer to my friend's Lieven great blog. However I disagree with your claim that bhwang must give credit to it "for bringing this passage to your mind". I have no idea what is going on in bhwang's mind and I believe that, indeed, he copied the passage from Lieven's blog. But that proves nothing: If I have a book and I want to reproduce a few paragraphs from it, I use Google to find a source online and then copy the needed lines from it without giving much attention to its environment.(to be continued) $\endgroup$ – Georges Elencwajg Dec 31 '10 at 11:25
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    $\begingroup$ (Continuation) Moreover, since MathOverflow is not a research online journal, specialists usually answer questions without giving references to original documents nor citing the source of their inspiration and I think this is the right attitude on our site. $\endgroup$ – Georges Elencwajg Dec 31 '10 at 11:29
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    $\begingroup$ I voted up Chandan Singh Dalawat's quotation as close to what I had in mind. But I didn't mark it as my accepted answer, because I remember reading somewhere a much shorter statement (by or about Weil) that explicitly mentions prime numbers. $\endgroup$ – Jonathan Sondow Jan 1 '11 at 14:42

Summary. Search for "Weil formules explicites".

Jonathan says in a comment that he is looking for a statement by Weil "that explicitly mentions prime numbers". This reminds me of his paper Sur les "formules explicites" de la théorie des nombres premiers. Comm. Sém. Math. Univ. Lund (1952). Tome Supplementaire, 252–265.

The review of this paper in Math Reviews says :

The most striking result of the paper is as follows. The author defines a distribution (too complicated to define here) whose positivity is equivalent to the simultaneous truth of the Riemann hypothesis for the Artin-Hecke $L$-series and the Artin conjecture on their entirety. This situation is analogous to the case of curves over finite fields for which the Riemann hypothesis is a consequence of the positivity of the trace in the ring of correspondences.

Let me also mention a paper by Burnol in the Comptes Rendus, of which the review says

As is known, the proof of A. Weil of the analog for algebraic curves of the Riemann hypothesis (R.H.) relies upon the equivalence of this hypothesis with the positivity of a suitable Hermitian form. Weil, again, remarked that also the original R.H. for $L(s,\chi)$ (the $L$-function associated to the Dirichlet character $\chi$) holds if and only if $Z(g\ast > g^\tau)=\sum_{\rho}\widehat{g}(\rho)\overline{\widehat{g}(\overline{1-\rho})}\geq > 0$ for every smooth compactly supported $g$, where $\rho$ runs over the critical zeros of $L(s,\chi),\ > \widehat{g}$ is the Mellin transform of $g$ and $g^\tau(u)=\overline{u^{-1}g(u^{-1})}$.

Addendum. It goes without saying that one should also read Weil's own commentary on his paper in vol. II of his Collected Papers.

  • $\begingroup$ Although the title of Weil's paper includes ``nombres premiers,'' the quoted reviews of his and Burnol's papers do not mention primes. So we still have no example of a quotation that explicitly links Weil to both the Riemann Hypothesis and prime numbers. $\endgroup$ – Jonathan Sondow Jan 9 '11 at 22:15
  • $\begingroup$ Geoffrey Caveney has pointed out a related statement by Brian Conrey: It is my belief that RH is a genuinely arithmetic question that likely will not succumb to methods of analysis." See the last paragraph of his article The Riemann Hypothesis'' in the March 2003 AMS Notices, available at ams.org/notices/200303/fea-conrey-web.pdf $\endgroup$ – Jonathan Sondow Oct 21 '11 at 20:46
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    $\begingroup$ In a 1978 interview in "Pour la Science" Weil said, "Or l’hypothèse de Riemann n’est pas un point isolé des mathématiques, mais au contraire, constitue un verrou de la théorie des nombres. Pour la démontrer, il faudrait d’abord mieux connaître, et par conséquent faire progresser la théorie des nombres." This is quoted by Michèle Audin on p. 666 of her book "Correspondance entre Henri Cartan et André Weil (1928-1991)", Société mathématique de France, 2011, available at ams.org/bookstore-getitem/item=SMFDM-6 for purchase. $\endgroup$ – Jonathan Sondow Jan 18 '12 at 15:54

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