A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line

Abstract: We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. A proof of these results was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by Alpöge and Furman. This argument is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank-trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new proof is shorter and proceeds by replacing the entire finite-dimensional matrix framework by a Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

• The paper proves that more than 67.25% of the non-trivial zeros of the Riemann zeta function are both simple and located on the critical line.
• The authors use an unconditional pair-correlation theorem and a Hilbert-space inequality to achieve these bounds, avoiding the need for any unproven hypotheses.
• The method not only proves the desired bounds but also simplifies and clarifies the mathematical approach compared to previous technical matrix arguments.

Problem setting and principal claims

The paper gives a new unconditional proof that a positive proportion exceeding two thirds of the non-trivial zeros of the Riemann zeta function are both simple and located on the critical line. Specifically, with the standard notation $N(T)$ for the number of non-trivial zeros with $0<\operatorname{Im}(\rho)\leq T$, counted with multiplicity, $N_0^s(T)$ for the number of simple critical-line zeros, and $N_d(T)$ for the number of distinct zeros, it proves

$$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$

and

$$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$

Thus, more than $67.25\%$ of the zeros are simple and on the critical line, while more than $83.62\%$ are distinct. The argument is unconditional in the sense that it does not assume RH, GLH, or a zero-density hypothesis. Its analytic input is an unconditional pair-correlation theorem of Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST24], while its principal new ingredient is a Hilbert-space inequality that replaces the finite-dimensional matrix construction used in the proof announced by Alpöge and Furman (Alpöge et al., 13 Aug 2026). The paper itself is identified as (2609.02882).

The result concerns two logically separate features of the zero set. A zero contributes to $N(T)$ according to its multiplicity, whereas $N_d(T)$ counts each distinct ordinate and real part only once. The simultaneous lower bound for $0<\operatorname{Im}(\rho)\leq T$0 is stronger than a statement merely about critical-line zeros or merely about simple zeros: it controls simplicity and horizontal location through one quadratic-form estimate.

Relation to earlier methods

The classical route to proportions of critical-line zeros is based on mollification. Selberg first established a positive proportion, Levinson exceeded one third, and Conrey obtained more than two fifths of the zeros as simple critical-line zeros. The work of Pratt, Robles, Zaharescu, and Zeindler later improved the corresponding unconditional critical-line and simple-critical-line estimates to approximately $0<\operatorname{Im}(\rho)\leq T$1 and $0<\operatorname{Im}(\rho)\leq T$2, respectively.

A different mechanism arises from Montgomery’s pair-correlation method. Under RH, Montgomery proved that at least $0<\operatorname{Im}(\rho)\leq T$3 of the zeros are simple, and Montgomery–Taylor optimization produced the constant

$0<\operatorname{Im}(\rho)\leq T$4

Subsequent work improved some RH-conditional bounds, but the relevant difficulty was to obtain comparable information without assuming that all zeros lie on the critical line. The unconditional pair-correlation theorem of Baluyot et al. [BGST24] supplies an asymptotic formula involving the complex differences $0<\operatorname{Im}(\rho)\leq T$5 and therefore permits off-line zeros. The present paper shows that, when combined with the appropriate Hilbert-space inequality, this input yields exactly the Montgomery–Taylor constant without any prior restriction on the real parts of the zeros.

The paper emphasizes that the obstacle is not simply the choice of a better positive kernel. The elementary multiplicity argument used in Montgomery’s original proof requires positivity of individual off-diagonal terms after the zeros are rescaled. For zeros off the critical line, this would effectively require a nonconstant entire kernel with nonnegative real part throughout the complex plane, which is impossible. The new argument avoids termwise positivity altogether.

The Hilbert-space inequality

The central finite-multiset result applies to any nonempty finite multiset $0<\operatorname{Im}(\rho)\leq T$6 invariant under complex conjugation. Let $0<\operatorname{Im}(\rho)\leq T$7 be a real-valued, even function supported in $0<\operatorname{Im}(\rho)\leq T$8, with $0<\operatorname{Im}(\rho)\leq T$9, and define

$N_0^s(T)$0

The proposition establishes the two inequalities

$N_0^s(T)$1

and

$N_0^s(T)$2

Multiplicity is retained in the double sum and in $N_0^s(T)$3, while the left-hand side of the second inequality counts distinct elements. These formulas reduce the problem to controlling a single quadratic pair-correlation expression.

The proof constructs, for each $N_0^s(T)$4, the function

$N_0^s(T)$5

Conjugation symmetry is encoded by decomposing $N_0^s(T)$6 into components invariant and anti-invariant under $N_0^s(T)$7:

$N_0^s(T)$8

The kernel admits the Gram-type factorization

$N_0^s(T)$9

This identity converts the quadratic sum of squared kernel values into the squared norm of a two-variable function,

$N_d(T)$0

namely

$N_d(T)$1

The key point is that the right-hand side is manifestly nonnegative even though individual terms $N_d(T)$2 need not be. The proof then introduces nested subspaces $N_d(T)$3 of $N_d(T)$4, generated respectively by the components associated with multiple real elements, all real elements, and both real and non-real conjugate pairs.

An orthonormal basis adapted to this nesting is obtained by Gram–Schmidt. The symmetry relation satisfied by the generating functions ensures that all relevant coefficients are real. The rank-like information distinguishing simple real elements and distinct elements is extracted by applying elementary scalar inequalities to the coefficients $N_d(T)$5 of $N_d(T)$6 against the tensor-product basis functions $N_d(T)$7. Bessel’s inequality then bounds the coefficient square sum by $N_d(T)$8.

For simple real elements, the three basis ranges yield a combined inequality of the form

$N_d(T)$9

Rearranging and invoking the kernel factorization gives the first assertion. A modified treatment of the same ranges yields the distinct-element inequality. The derivation is conceptually important: simplicity and distinctness emerge from a single Hilbert-space second moment rather than from a matrix rank–trace argument.

Application to zeta zeros

For the zeta function, the paper applies the proposition to the multiset

$$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$0

with each zero repeated according to multiplicity. The functional equation maps $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$1 to $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$2, which becomes complex conjugation on $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$3. Consequently, an element of $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$4 is real exactly when $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$5, and its multiplicity is the multiplicity of the associated zeta zero. The abstract multiset inequalities therefore become

$$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$6

and

$$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$7

The remaining task is to produce a kernel for which the pair-correlation sum is asymptotic to the Montgomery–Taylor constant times $$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$8.

Removing the pair-correlation weight

The unconditional pair-correlation formula in [BGST24] applies to sums of the form

$$\liminf_{T\to\infty}\frac{N_0^s(T)}{N(T)} \geq C_0 =0.6725007\ldots,$$9

and yields

$$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$0

The Hilbert-space proposition, however, requires an unweighted sum involving $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$1. Directly absorbing $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$2 into a test function would make the test function depend on $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$3, preventing a direct application of the pair-correlation theorem. The paper resolves this mismatch by a fixed linear-combination construction.

Choose a smooth, real-valued, even function $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$4 supported in $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$5, normalized so that $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$6, and set

$$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$7

Then $$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$8. Define

$$\liminf_{T\to\infty}\frac{N_d(T)}{N(T)} \geq \frac{1+C_0}{2} =0.8362503\ldots.$$9

Twice integrating by parts gives

$67.25\%$0

At the scaled zero differences, this factor is the reciprocal of the pair-correlation weight:

$67.25\%$1

Multiplication by $67.25\%$2 therefore recovers the unweighted kernel square. Since $67.25\%$3 is a linear combination of the two fixed admissible test functions $67.25\%$4 and $67.25\%$5, the unconditional pair-correlation formula applies separately to both terms. The derivative contribution is suppressed by $67.25\%$6, while the error remains $67.25\%$7.

The resulting asymptotic is

$67.25\%$8

where

$67.25\%$9

Extremal optimization and numerical bounds

The test functions are chosen to approximate the Montgomery–Taylor extremizer. The limiting profile is

$83.62\%$0

which is positive, even, and normalized to have integral one. Smooth compactly supported approximations $83.62\%$1 converge to $83.62\%$2 in $83.62\%$3, and hence $83.62\%$4 converges to

$83.62\%$5

The extremal result of Carneiro, Chandee, Littmann, and Milinovich shows that this constant is optimal within the relevant class of pair-correlation test functions (Davidson et al., 2017). Thus the constant is not an artifact of the smoothing procedure: it is the sharp value accessible through this quadratic-form method and the stated support constraint.

Substitution into the multiset inequalities gives

$83.62\%$6

and

$83.62\%$7

The second constant is algebraically related to the first by

$83.62\%$8

Hence the distinct-zero estimate is not obtained through an independent analytic optimization; it is a second consequence of the same pair-correlation bound.

Limitations and open questions

The proof depends essentially on the unconditional pair-correlation formula of [BGST24], including its support restriction and its stated uniformity. The paper does not derive that theorem, so the main result is conditional on the validity of that external analytic input, although no RH-type hypothesis is imposed.

The argument also reaches the Montgomery–Taylor constant but does not exceed it. The cited extremal theorem shows that this limitation is intrinsic to the present kernel optimization, not merely to the particular smooth approximation selected in the proof. Improving the numerical proportion would therefore require either a stronger analytic pair-correlation estimate, a different quadratic inequality, or information beyond the class of test functions used here.

The Hilbert-space proposition is formulated for finite conjugation-invariant multisets and is exact at that level. Passing to zeta zeros introduces the asymptotic regime, the truncation at height $83.62\%$9, and the pair-correlation error term. The proof consequently establishes liminf bounds rather than an effective finite-$N(T)$0 proportion. It also leaves open whether a comparable Hilbert-space mechanism can exploit finer horizontal information about off-line zeros or yield bounds beyond the Montgomery–Taylor extremal barrier.

The appendix reports machine-generated Lean certificates for the finite-multiset proposition and for the theorem after importing the pair-correlation lemma and the Riemann–von Mangoldt asymptotic. These certificates provide formal verification of the encoded statements, but they do not formalize the external analytic theorem of [BGST24] within the paper.

Conclusion

The paper replaces a technically elaborate finite-dimensional matrix argument with a direct Hilbert-space inequality for conjugation-invariant multisets. This reformulation converts the simultaneous problem of simplicity and critical-line location into the estimation of one squared-kernel pair-correlation sum. The unconditional pair-correlation theorem of [BGST24], combined with a differential correction that removes its rational weight, yields the sharp Montgomery–Taylor constant and consequently the bounds $N(T)$1 for simple critical-line zeros and $N(T)$2 for distinct zeros. The method is shorter and structurally more transparent, while its numerical limitation is explicitly tied to the underlying extremal problem.