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.

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.

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 $g^\tau(u)=\overline{u^{-1}g(u^{-1})}$.

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.