First-hand information for everyone
This paper gives a new, simpler proof that more than 67.25% of the non-trivial zeros of the Riemann zeta function are simple and lie on the critical line. Here a “zero” means a place where the zeta function equals zero, the “critical line” means the vertical line in the complex plane with real part 1/2, and “simple” means the zero has multiplicity one (it is not a repeated root). The author also shows that at least 83.62% of the non-trivial zeros are distinct from one another.
Instead of the earlier, more intricate argument recently produced by an internal research version of the AI Claude and verified by Alpöge and Furman, the author replaces a finite-dimensional matrix construction with a single inequality in an infinite-dimensional space (a Hilbert space). This change removes most of the linear algebra bookkeeping. The remaining step is to control a certain quadratic expression that measures how the zeros are spaced.
A key tool is Montgomery’s pair-correlation idea. Pair correlation is a way to study how often pairs of zeros occur at a given spacing; it is a statistical measure of the zeros’ spacing. The paper uses an unconditional version of Montgomery’s pair-correlation theorem proved by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. That input lets the author estimate the needed quadratic form and get the numerical proportions above.
Why this matters: showing large fractions of zeros lie on the critical line is progress toward the Riemann hypothesis, which predicts that all non-trivial zeros lie there. These results do not prove the Riemann hypothesis, but they give stronger, unconditional evidence about where most zeros are and how they behave. Improving the conceptual simplicity of the argument also makes the underlying ideas clearer and easier to check.
Important caveats: the result is not a proof of the Riemann hypothesis and covers a proportion, not all, of the zeros. The numerical bounds come from optimizing a specific extremal problem tied to Montgomery’s method, so the same numerical constants appear here and in the earlier Claude-based argument. The proof depends on the recent unconditional pair-correlation estimates of Baluyot et al., and the paper notes a technical step where the test function must be adjusted to remove a weight that appears in that unconditional formula.
The paper includes further technical details, references to the long history of related work, and formal verification notes; the author mentions Lean formal certificates produced with help from the Axiom Math team. For readers curious about the technical steps, the full write-up gives the precise inequalities and the analytic estimates behind the proportions.