Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory rather than by analysis?

Question

Did André 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?

Answer 1

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 exemple, ou pour mieux dire le paradigme, des lois dites “du corps de classe”, qui gouvernent les extensions abéliennes des corps de nombres 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 symétries 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 exemple 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’hypothèse 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 (pages 3 and 4 of 12).

"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 not 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."

Answer 2

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.

Answer 3

I would like to bring your attention to the paper by Andre Weil titled "Two lectures on number theory, past and present". This is based on a talk he gave At Columbia University in 1972.

His term "number theory" stands for, in my opinion, what we now called "the elementary number theory" and "the algebraic number theory".

His term "analysis" stands for, in my opinion, what we now called "the analytical number theory".

But I was not able to find the sentence that implied that Riemann Hypothesis (for number field) would be settled by (prime) number theory than by analysis.