On the OpenAI proof that Landau-Siegel zeros do not exist

Abstract: A recent manuscript of OpenAI asserts that Landau-Siegel zeros do not exist. Here, we give a detailed account of its argument, arranged as a direct proof with explicit constants. There are absolute constants $q_0$ and $δ_0>0$ such that, for every primitive real Dirichlet character $χ$ of conductor $q\ge q_0$, every real zero $β\in(0,1)$ of $L(s,χ)$ satisfies $(1-β)\log q\geδ_0$, and one can take $q_0=10^{10}$ and $δ_0=3.0\times 10^{-3}$. Combined with a recent computation of Lu, Zaman, and Zhao, this result yields the bound $(1-β)\log q\ge 3.0\times 10^{-3}$ for every conductor $q\ge 3$. The proof is an interpolation determinant argument in the biquadratic field generated by $\sqrt{2}$ over the quadratic field attached to $χ$. A real zero close to one forces $χ(p)=-1$ for most primes $p$, and the Frobenius congruence at these primes makes a certain nonzero determinant divisible by higher powers of these primes than its Archimedean size allows. The underlying ideas are those of the OpenAI manuscript. The explicit constants, and the presentation, are new.

• The paper **provides a detailed, explicit method to prove Landau-Siegel zeros do not exist for real Dirichlet L-functions**, building on an argument attributed to an OpenAI manuscript with key improvements in explicating parameters and inserting specific constants.
• The conclusion establishes a universal lower bound for the imaginary part of real zeros, demonstrating that $δ = (1-β) log q ≥ 3.0 × 10−3$ for all conductors $q ≥ 3$ after adjusting for an explicit computation based on an external paper.
• The diminished presence of negative principal characteristic primes under a zero near 1 mandates a decisive bias, reinforcing OpenAI's initial linkages between dirichlet determinants and primes.

Problem setting and principal claim

The paper studies the existence of Landau–Siegel zeros for primitive real Dirichlet $L$-functions. If $\chi$ is a primitive real character of conductor $q$ and $L(\beta,\chi)=0$ for a real $\beta$ close to $1$, the conventional parameter is

$\delta=(1-\beta)\log q.$

A Landau–Siegel zero would correspond to $\delta$ being exceptionally small. The paper claims an effective absolute lower bound

$(1-\beta)\log q\ge 3.0\times 10^{-3}$

for every real zero and every conductor $q\ge 3$, after combining its large-conductor argument with a computation attributed to Lu, Zaman, and Zhao (Lu et al., 3 Feb 2026).

The main theorem proved in the paper is stated initially for $\chi$0: every real zero of $\chi$1 satisfies

$\chi$2

The extension to $\chi$3 is not derived analytically in the paper. Instead, it invokes a finite computation establishing the stronger exclusion $\chi$4 for those conductors. Thus the claimed universal constant depends on both the explicit determinant argument and the correctness and scope of the external computation.

The result is presented as a detailed and explicit version of an argument attributed to an OpenAI manuscript, rather than as an original conceptual method. The paper’s contribution is the direct organization of the argument, the introduction of explicit parameter choices, and the numerical constants.

Analytic reduction to a bias among primes

The only point at which the putative zero $\chi$5 enters the proof is an upper bound for the logarithmic derivative $\chi$6 for real $\chi$7. Starting from the partial-fraction expansion of $\chi$8, the contribution of the zero $\chi$9 is isolated, while the contributions of all other zeros are discarded using their nonnegative real parts. For $q$0, this gives an inequality of the form

$q$1

with an explicit positive constant $q$2.

Combining this estimate with the corresponding bound for $q$3 yields an inequality for

$q$4

Choosing $q$5 and retaining only prime terms produces an upper bound for the weighted contribution of primes satisfying $q$6. After an explicit Mertens estimate, the paper obtains

$q$7

For $q$8, the paper takes

$q$9

Consequently, if $L(\beta,\chi)=0$0 is small and $L(\beta,\chi)=0$1 is a sufficiently large power of $L(\beta,\chi)=0$2, then primes with $L(\beta,\chi)=0$3 carry nearly all of the $L(\beta,\chi)=0$4 mass up to $L(\beta,\chi)=0$5. This is the analytic manifestation of the exceptional-character prime bias: a zero close to $L(\beta,\chi)=0$6 forces the character to take the value $L(\beta,\chi)=0$7 on most primes in the relevant weighted sense.

The implication for the determinant argument is specific and essential. Only primes with $L(\beta,\chi)=0$8 generate the Frobenius automorphisms needed to lower the determinant’s monomial weight. The proof therefore requires not merely a general prime bias, but concentration on precisely the Frobenius classes compatible with the chosen interpolation variables.

The interpolation construction

The algebraic component is built in the biquadratic field

$L(\beta,\chi)=0$9

where $\beta$0 is the quadratic field associated with $\beta$1. Writing $\beta$2 and $\beta$3, the relevant algebraic integers are

$\beta$4

The proof uses the three conjugates

$\beta$5

where $\beta$6 changes the sign of $\beta$7 and $\beta$8 changes the sign of $\beta$9. These define a linear map from $1$0 to $1$1 with one-dimensional kernel. The four integer coordinates are linearly independent over $1$2, which ensures that the $1$3 lattice points remain distinct under this projection.

The central interpolation proposition asserts that arbitrary data on these $1$4 points can be interpolated by a polynomial in three variables whose coordinatewise degrees are only of order $1$5. More precisely, if the degree parameters $1$6 satisfy

$1$7

and

$1$8

then every function on the projected lattice set is represented by a polynomial with coordinatewise degrees bounded by the $1$9.

The proof is not ordinary tensor-product interpolation. It uses a dimension argument, a convex-geometric supporting-face construction, and a nearest lattice point to a dilated convex hull. An auxiliary polynomial vanishes on all lattice points in the dilated polytope. A nearest lattice point where this auxiliary polynomial does not vanish yields a supporting face and a distinguished point $\delta=(1-\beta)\log q.$0 in the support of a hypothetical linear dependence. A further polynomial isolates $\delta=(1-\beta)\log q.$1 from the other points. The resulting contradiction establishes surjectivity of the evaluation map.

This proposition yields a particularly important anisotropic spanning statement. Setting

$\delta=(1-\beta)\log q.$2

the monomial rows with degree bounds

$\delta=(1-\beta)\log q.$3

span the full $\delta=(1-\beta)\log q.$4-dimensional function space. The asymmetry is later exploited to force most of the total degree into the first variable.

The weighted determinant

For $\delta=(1-\beta)\log q.$5, define the row

$\delta=(1-\beta)\log q.$6

The rows are ordered by the weight

$\delta=(1-\beta)\log q.$7

A greedy procedure retains a row exactly when it is linearly independent of all preceding retained rows. The interpolation result guarantees that the retained rows form a basis of the $\delta=(1-\beta)\log q.$8-dimensional space, and hence define a nonzero determinant

$\delta=(1-\beta)\log q.$9

Let

$\delta$0

where the sums range over retained rows. The paper proves that, for its eventual choice of $\delta$1,

$\delta$2

Thus, most of the determinant’s degree is concentrated in the first conjugate variable. This is the structural reason the $\delta$3-adic argument can produce a lower bound with a leading coefficient $\delta$4, whereas the Archimedean estimate has leading coefficient only approximately $\delta$5.

The rank discrepancy is decisive: there are four lattice coordinates but only three interpolation variables. Since $\delta$6, one has

$\delta$7

The proof’s contradiction is ultimately generated by this factor $\delta$8. The auxiliary quadratic extension by $\delta$9 is therefore not cosmetic; it increases the lattice rank without introducing a fourth interpolation variable.

Archimedean upper bound

Every Galois conjugate of $(1-\beta)\log q\ge 3.0\times 10^{-3}$0 has size bounded by

$(1-\beta)\log q\ge 3.0\times 10^{-3}$1

Applying Hadamard’s inequality to each Galois conjugate of the determinant gives

$(1-\beta)\log q\ge 3.0\times 10^{-3}$2

Using $(1-\beta)\log q\ge 3.0\times 10^{-3}$3 and $(1-\beta)\log q\ge 3.0\times 10^{-3}$4, the main term is bounded by

$(1-\beta)\log q\ge 3.0\times 10^{-3}$5

The terms involving $(1-\beta)\log q\ge 3.0\times 10^{-3}$6, $(1-\beta)\log q\ge 3.0\times 10^{-3}$7, and $(1-\beta)\log q\ge 3.0\times 10^{-3}$8 are controlled by taking $(1-\beta)\log q\ge 3.0\times 10^{-3}$9 to be a sufficiently large fixed power of $q\ge 3$0.

This estimate is an upper bound on the size of a nonzero rational integer, namely $q\ge 3$1. The proof can succeed only if the arithmetic divisibility supplied by the admissible primes forces a strictly larger lower bound.

Frobenius congruences and the $q\ge 3$2-adic lower bound

For an odd prime $q\ge 3$3 with $q\ge 3$4, the Frobenius action on $q\ge 3$5 is either $q\ge 3$6 or $q\ge 3$7, depending on the value of the quadratic character of $q\ge 3$8 modulo $q\ge 3$9. In the ring $\chi$00, the congruence

$\chi$01

holds for every $\chi$02, where $\chi$03 is one of these two automorphisms.

The choice of interpolation variables is tailored to this congruence: $\chi$04 is one of the three coordinates represented in the monomial rows. If

$\chi$05

then the first factor in a row can be replaced by

$\chi$06

Every entry of the resulting row is divisible by $\chi$07. At the same time, expanding the replacement expresses the new row as the original row plus rows of strictly smaller weight, provided $\chi$08. The determinant is therefore unchanged under this lower-triangular row operation, while acquiring a factor

$\chi$09

from each retained row.

Consequently,

$\chi$10

and hence

$\chi$11

Summing over admissible primes $\chi$12 with $\chi$13, and using

$\chi$14

gives

$\chi$15

The prime-bias estimate then yields

$\chi$16

When $\chi$17 is small, the leading term is approximately $\chi$18, which exceeds the Archimedean leading term at most $\chi$19. This is the central incompatibility: the determinant is forced to be divisible by too large an integer to have the asserted Archimedean size.

Explicit parameter choices and numerical bound

The paper chooses

$\chi$20

with $\chi$21 after substituting the explicit constants

$\chi$22

For $\chi$23, the error terms are absorbed into the main comparison. Combining the Archimedean and $\chi$24-adic inequalities gives

$\chi$25

Since $\chi$26 is comparable to $\chi$27, the second term is made sufficiently small by the choice $\chi$28. The resulting lower bound is

$\chi$29

The paper rounds this down to

$\chi$30

The numerical result is strong in the sense that it is fully effective for all $\chi$31, rather than merely asserting the existence of an unspecified absolute constant. However, the size of the constant is highly sensitive to the deliberately conservative interpolation and determinant estimates. The argument establishes a positive explicit constant, but does not claim numerical optimality.

Limitations and open questions

The principal claim is exceptionally strong: a uniform effective exclusion of Landau–Siegel zeros would resolve a longstanding problem in analytic number theory. The paper therefore requires scrutiny at a level beyond checking the broad strategy. Its proof depends on several delicate components simultaneously: the interpolation proposition, the greedy basis construction, the extraction of prime powers under weighted row operations, the explicit prime-bias inequality, and the final constant comparison. A defect in any one of these steps would invalidate the theorem.

The manuscript itself identifies the computation for conductors below $\chi$32 as an external input. The analytic argument proves the theorem only for large conductors; the universal bound $\chi$33 additionally depends on the cited computation (Lu et al., 3 Feb 2026), which is not reproduced or independently verified within the paper. The paper also relies on the assertion that the OpenAI manuscript contains the underlying valid argument, while its own originality is restricted to exposition and explicit constants.

Several auxiliary statements are presented tersely. In particular, the nonvanishing lemma for a nonzero polynomial on $\chi$34 is labeled an exercise, even though it is used in the interpolation proof. This omission is easily repairable, but the paper’s central result requires every such auxiliary claim to be integrated into a formally complete verification. More generally, the interpolation proposition is unusually strong and is the technical hinge connecting the three-variable determinant to the four-dimensional lattice; its precise hypotheses and all field-of-definition issues require independent checking.

The main open question left by the paper is therefore not a qualitative one about how the determinant mechanism would generalize, but a concrete verification question: does every step of the interpolation-and-divisibility argument remain valid with the stated constants and with no unaddressed dependence on $\chi$35, $\chi$36, or the choice of retained rows? If so, the claimed theorem would provide an effective exclusion of Landau–Siegel zeros; if not, the failure would have to occur in one of the explicit linear-algebraic, algebraic-number-theoretic, or analytic inequalities identified above.

Conclusion

The paper presents a determinant proof of a uniform lower bound for real zeros of primitive real Dirichlet $\chi$37-functions. Its mechanism combines an analytic prime bias caused by a hypothetical zero near $\chi$38, a three-variable interpolation scheme on a four-dimensional lattice, Frobenius congruences in a biquadratic extension, and incompatible Archimedean and $\chi$39-adic bounds for a nonzero algebraic determinant. With explicit parameter choices, it claims

$\chi$40

for all sufficiently large conductors and, using an external computation, for all $\chi$41. The paper’s significance rests entirely on the correctness of this tightly coupled chain of explicit estimates and algebraic reductions (2610.15483).