Convergence of zeros from finite-prime Weil minimizers to the zeta zeros

Show that, as the upper cutoff x increases (equivalently, as the finite set of primes used in the Weil quadratic form expands), the zeros of the Mellin transform of the minimizer η_x of the restricted Weil quadratic form converge to the nontrivial zeros of the Riemann zeta function.

Background

In the letter, the author describes an optimization procedure that constructs, for a finite set of primes up to x, a quadratic form whose minimizer η_x yields a Mellin transform with all zeros on the critical line. Numerically, using primes up to 13, this produces striking approximations to the first 50 zeros.

The unresolved issue is whether, as x → ∞, the zeros of these approximating functions actually converge to the true nontrivial zeros of ζ(s). Establishing such convergence would provide a path to proving the Riemann Hypothesis via Hurwitz’s theorem.

References

What we do not know is that, when we increase the upper limit, which was x=13 here, the corresponding set of zeros will converge towards the zeros of zeta. This is something which at this point is not proved.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in A Letter to Professor Bernhard Riemann

Relating this residual quantitatively to explicit prime and prime-power contributions remains an open problem.

— Hidden Dyson Universality in Inverse-Spectral Geometry  (2608.13475 - Hayashi et al., 13 Aug 2026) in Section “Riemann zeros and height dependence”

The central unresolved task is no longer the algebraic removal of a small ground state. It is a relative perturbation theorem comparing the entire arithmetic pencil with the exact finite-zero pencil in the quotient-energy norm. Such a theorem must control the positive zero tail relative to the compressed metric, preserve separated generalized eigenvalues and multiplicities, and lead to a limiting operator or determinant without spectral pollution.

— Construction of Finite Hilbert--Pólya Matrices from Weil's Explicit Formula  (2609.04908 - Shi, 4 Sep 2026) in Section 1, subsection “Arithmetic approximation and the remaining transfer problem”