More than two thirds of the zeta zeros are simple and on the critical line

Technical Summary: "More Than Two Thirds of the Zeros of the Riemann Zeta Function Are Simple and on the Critical Line"

Problem Statement
The paper addresses the distribution of nontrivial zeros of the Riemann zeta function, $\zeta(s)$. Specifically, it seeks unconditional lower bounds for two quantities as $T \to \infty$:

1. The proportion of zeros lying on the critical line ($\text{Re}(s) = 1/2$) that are simple (multiplicity 1).
2. The proportion of distinct zeros (counting multiplicity as 1 regardless of order) lying on the critical line.

Prior to this work, the unconditional records were $5/12$ ($\approx 0.4167$) for simple zeros on the line and $0.6603$ for distinct zeros. Under the Riemann Hypothesis (RH), Montgomery (1973) deduced that at least $2/3$ of zeros are simple on the line, and subsequent work improved distinct zero bounds. The challenge has been to remove the reliance on RH, which was historically required to ensure the positivity of the "zero side" of Weil's explicit formula when summing over real ordinates.

Methodology
The proof constructs a finite-dimensional approximation of Weil's explicit formula and applies linear algebraic rank-trace inequalities to bypass the need for RH.

1. Weil's Explicit Formula and Hermitian Forms:
   The authors utilize Weil's explicit formula to define a Hermitian form $W(f, g) = \sum_\rho m_\rho \hat{f}(\gamma_\rho) \overline{\hat{g}(\gamma_\rho)}$ on test functions. The positivity of this form on all compactly supported functions is equivalent to RH.
   Instead of assuming positivity, the authors restrict this form to a specific family of $d \approx N(T, 2T)$ modulated test functions (a "window" $\psi$) spaced over the interval $[T, 2T]$. This yields a real symmetric $d \times d$ matrix $\tilde{G}$.

2. Matrix Decomposition (The Zero Side):
   The matrix $\tilde{G}$ is decomposed into $\tilde{G} = P + Q + E$, where:

   • $P$ represents contributions from zeros on the critical line.
   • $Q$ represents contributions from off-line pairs $\{\rho, 1-\bar{\rho}\}$.
   • $E$ is a tail term with small trace norm.

   Crucially, the authors analyze the rank and positive index (number of positive eigenvalues) of these components:

   • Each simple zero on the line contributes rank 1 to $P$.
   • Each off-line pair contributes a block of signature $(1, 1)$ to $Q$, meaning it contributes 1 to the positive index $n_+(Q)$ but 0 to the rank of the "on-line" part.
   • This structural insight, attributed to Bombieri, allows the authors to bound the number of off-line pairs using the positive index of $Q$.

3. Rank-Trace Inequality (The Core Innovation):
   The paper introduces a specific linear algebraic inequality (Lemma 3.2) for Hermitian matrices $P_1 \succeq 0$ and $Q'$:
   $$\text{rank}(P_1) \ge 2\text{tr}(P_1) + 4\text{tr}(Q') - 4n_+(Q') - \|P_1 + Q'\|_{HS}^2$$
   By setting $P_1$ as the part of $P$ corresponding to simple on-line zeros and $Q'$ as the remainder, the authors derive a lower bound for the number of simple on-line zeros ($N_s^0$) purely in terms of the trace and Frobenius norm of $\tilde{G}$ and the count of off-line pairs. This replaces the classical requirement of termwise positivity (which fails off the line) with a "rank-trace" bound.

4. The Prime Side (Unconditional Evaluation):
   The trace $\text{tr}(\tilde{G})$ and the squared Frobenius norm $\|\tilde{G}\|_{HS}^2$ are evaluated using the "prime side" of Weil's formula.

   • $\text{tr}(\tilde{G})$ is shown to be asymptotically $N(T, 2T)$.
   • $\|\tilde{G}\|_{HS}^2$ is evaluated as a mean value of a Dirichlet polynomial. This relies on the Montgomery–Vaughan inequality and bounds on sums of $\Lambda(n)^2$.
   • The result is $\|\tilde{G}\|_{HS}^2 = (R(\psi) + o(1))N(T, 2T)$, where $R(\psi)$ is a constant depending on the window function. For the indicator window, $R(\psi_0) = 4/3$.

Key Contributions and Results
The paper establishes the following unconditional results for the interval $(T, 2T)$:

• Theorem A(i): At least $2/3$ (specifically $2/3 - o(1)$) of the nontrivial zeros are simple and lie on the critical line.
  • Using the Montgomery–Taylor window, this constant improves to $0.6725$.
• Theorem A(ii): At least $5/6$ (specifically $5/6 - o(1)$) of the nontrivial zeros are distinct and lie on the critical line.
  • Using the Montgomery–Taylor window, this constant improves to $0.8362$.
• Theorem B: These results extend verbatim to primitive Dirichlet $L$-functions $L(s, \chi)$.

The proof does not use mollifiers, zero-density estimates, or zero-free regions. It relies on the explicit formula, Stirling's estimate, Chebyshev–Mertens estimates, and the Montgomery–Vaughan inequality.

Significance and Claims

• Making Montgomery's Deduction Unconditional: The paper claims to make Montgomery's 1973 deduction of the $2/3$ proportion unconditional. Classically, RH was needed to interpret the zero-side sum as a positive sum over real ordinates. Here, the authors replace this positivity with a rank-trace inequality applied to a finite compression of Weil's form.
• Optimality of the Method: The authors argue that the $2/3$ bound is sharp for this specific method. They demonstrate that the inequality used is sharp given only the trace, Frobenius norm, and block structure of the matrix. To improve the bound beyond $2/3$ using this specific "rank-trace" approach would require pair-correlation information beyond Fourier support 1 (i.e., knowledge of higher moments of the prime side), which is currently unavailable unconditionally.
• Formal Verification: The mathematical argument was discovered and written by an AI (Claude), verified by human authors, and formally verified in Lean 4. The formalization confirms that the proof relies only on standard axioms.
• Limitations: The paper explicitly states what the results are not. They are lower bounds only; the remaining zeros are not proven to be off the line, merely not captured by this specific certificate. The method is insensitive to $o(N)$ off-line zeros and would hold even for functions where the Riemann Hypothesis is false (e.g., Davenport–Heilbronn zeta functions), provided the arithmetic inputs hold.

In summary, the paper provides a rigorous, unconditional proof that a significant majority ($>66\%$) of zeta zeros are simple and on the critical line, achieving a bound that matches the classical conditional result of Montgomery through a novel application of linear algebraic rank constraints to the explicit formula.