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.