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Math-77 · mathematics · Number theory

Siegel Zeros

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A primitive quadratic character $\chi$ has minimal modulus, its conductor $q$, and values in $\{0,\pm1\}$. Using its Dirichlet $L$-function, the uniform zero-free assertion is \[ \exists\delta>0\ \forall q\ge3\ \forall\chi\text{ primitive quadratic of conductor }q, \quad L(\beta,\chi)\ne0\quad\left(1-\delta/\log q<\beta<1\right). \] The same $\delta$ must work for all conductors, and only real $\beta$ are quantified here. A separate positive zero-free radius for each individual character would not establish the target.

Formulation and concept references: [S1].

Short English statement

Can one uniformly exclude exceptionally close real zeros of quadratic Dirichlet $L$-functions near one?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Math-77-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-77

Theorem 1 of the uniform-exclusion manuscript separates real zeros from one by a common logarithmic conductor gap: $L(\beta,\chi)=0$, $\beta\in(0,1)$ implies $1-\beta\ge c/\log q$ for primitive real nonprincipal $\chi$ of conductor $q\ge3$, with one absolute $c>0$. This excludes the exceptional near-one zero phenomenon uniformly in the conductor. The other manuscripts additionally assert their specified fixed zero-free half-planes.

Siegel Zeros

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 003; 2026-10-05), the asserted result is that zeta, Dirichlet functions, and finite-order Eisenstein-field Hecke families admit a zero-free region with boundary $11/12$; In manuscript 2 (family 003; 2026-09-30), the asserted result is that zeta, Dirichlet $L$-functions, and finite-order Hecke $L$-functions for $\mathbb Q(\sqrt{-3})$ have no zeros with real part greater than $7/8$, with poles excluded; In manuscript 3 (family 003; 2026-10-01), the asserted result is that an absolute positive constant bounds $(1-\beta)\log q$ away from zero for real zeros of primitive nonprincipal real Dirichlet characters of conductor $q\ge3$.

Mathematical statement of the claimed result

Manuscript 1.
For $K=\mathbb Q(\sqrt{-3})$ and every finite-order Hecke character $\chi$ over $K$, the claimed zero-free assertion, with meromorphic poles excluded from evaluation, is \[ \operatorname{Re}s>11/12\quad\Longrightarrow\quad L_K(s,\chi)\ne0. \] It includes every Dirichlet $L$-function and the Riemann zeta function. The accompanying prime-progression claim uses $\pi(x;q,a)$ and asserts, with an absolute effective constant, \[ \sup_{\substack{1\le q\le x\$a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)}\int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x\quad(x\ge2). \] The critical-line requirement $\operatorname{Re}\rho=1/2$ is not a consequence of this half-plane claim.
Manuscript 2.
For every finite-order Hecke character of $F=\mathbb Q(\sqrt{-3})$, every Dirichlet character and the zeta function, the principal claim is \[ \operatorname{Re}s>7/8\quad\Longrightarrow\quad L(s)\ne0, \] where a principal character's pole at $s=1$ is allowed. The weaker displayed $11/12$ theorem follows within this scope. The cited arithmetic consequence gives absolute $A,C>0$ such that every odd prime $p$ has least positive quadratic nonresidue \[ n(p)\le C(\log p)^A. \] Thus $n(p)\ll_{\delta} p^{\delta}$ for each fixed $\delta>0$. Given binary $p$ and $a\in\mathbb F_p$, a deterministic algorithm of time polynomial in $\log p$ is further claimed to return a square root of $a$ or correctly declare it nonexistent. The half-plane assertion leaves zeros between the critical line and $7/8$ untreated.
Manuscript 3.
The manuscript's Theorem 1 asserts an absolute positive constant, uniform over all real primitive characters: \[ \exists c>0\ \forall q\ge3\ \forall\chi\text{ primitive, real, nonprincipal of conductor }q \ \forall\beta\in(0,1):\quad L(\beta,\chi)=0 \ \Longrightarrow\ (1-\beta)\log q\ge c. \] Thus these real zeros cannot approach one faster than a constant multiple of $1/\log q$. The assertion concerns real Dirichlet zeros; it does not place all nonreal zeros on the critical line.

Outline of the argument

For manuscript 1 (family 003; 2026-10-05), twisted Möbius sums are enlarged to a family before Poisson summation. A recursive estimate for the dual mean square, followed by inversion of cube factors, supplies the saving used to exclude a reciprocal-function singularity.

For manuscript 2 (family 003; 2026-09-30), smoothed character sums are estimated in two ways: directly by large-sieve and recursive moment bounds, and through a Mellin transform involving the reciprocal of an $L$-function. Poisson summation and sextic residue characters align the two expressions so that their savings force continuation of the reciprocal.

For manuscript 3 (family 003; 2026-10-01), a hypothetical exceptional zero forces a pronounced prime bias. Interpolation on a projected integer box produces a weighted conjugate determinant whose Archimedean size conflicts with its divisibility at primes selected by that bias.

Formal verification

For manuscript 2 (family 003; 2026-09-30), OpenAI lists Lean statements for this assertion: zeta, Dirichlet $L$-functions, and finite-order Hecke $L$-functions for $\mathbb Q(\sqrt{-3})$ have no zeros with real part greater than $7/8$, with poles excluded.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

For manuscript 3 (family 003; 2026-10-01), OpenAI lists Lean statements for this assertion: an absolute positive constant bounds $(1-\beta)\log q$ away from zero for real zeros of primitive nonprincipal real Dirichlet characters of conductor $q\ge3$.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

No Lean formalization is listed at this snapshot for manuscript 1 (family 003; 2026-10-05).

The linked family Comparator checks are DirichletSevenEighths.lean, HeckeSevenEighths.lean, QuasiRiemannHypothesis.lean, SiegelZeros.lean.

Their named declarations include OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SiegelZeros.WeightedTorusJets.dirichletRealZeroBound_proof, OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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Math-77-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-77

Siegel Zeros

Definitions and mathematical statement

A primitive quadratic character $\chi$ has minimal modulus, its conductor $q$, and values in $\{0,\pm1\}$. Using its Dirichlet $L$-function, the uniform zero-free assertion is \[ \exists\delta>0\ \forall q\ge3\ \forall\chi\text{ primitive quadratic of conductor }q, \quad L(\beta,\chi)\ne0\quad\left(1-\delta/\log q<\beta<1\right). \] The same $\delta$ must work for all conductors, and only real $\beta$ are quantified here. A separate positive zero-free radius for each individual character would not establish the target.

Formulation and concept references: [S1].

Short English statement

Can one uniformly exclude exceptionally close real zeros of quadratic Dirichlet $L$-functions near one?

Research attempt

Current frontier.
Lu--Zaman--Zhao report a rigorous computational exclusion of real zeros in $\sigma\ge1-1/(5\log q)$ for quadratic characters with $q\le10^{10}$ [S1], Theorem 1.1. Their verified finite range is not an all-conductor theorem. Their algorithm's asymptotic termination analysis uses GRH, separately from the finite verifications. The established zero-free-region theorem allows at most one exceptional real zero rather than excluding that zero. The distinction between lower bounds for $L(1,\chi)$ and a zero-free interval of width $1/\log q$ is analyzed by Friedlander--Iwaniec [E1], Sections 1--2. These are the relevant inputs; a general proof claim retained in [S2] is not used as a theorem.
Attack route.
Three possible routes are: control the variation of $L$ relative to its value at one; exploit positivity of the coefficients of $\zeta L$; or obtain a uniform lower bound for the positive prime sums used by the numerical exclusion method. The first requires an estimate proportional to $L(1,\chi)$, not just an absolute derivative estimate. The second must continue through a pole without illegally continuing a positive series. The third requires uniform information on splitting primes in a range where exceptional zeros are precisely an obstruction. We carry out the first route explicitly and test the positivity shortcut against a model.
Attempt: an unconditional derivative calculation.
Let $q\ge3$, $\ell=\log q>1$, and let $\chi$ be primitive quadratic. It is nonprincipal. Set $A(t)=\sum_{n\le t}\chi(n)$. Periodicity and the zero sum of a nonprincipal character over one full period give $A(q)=0$ and $|A(t)|\le q$. Partial summation yields, for real $s>0$, \[ L(s,\chi)=\sum_{n\le q}\frac{\chi(n)}{n^s} +s\int_q^\infty A(t)t^{-s-1}\,dt. \tag{43.1} \] The integral converges absolutely and agrees with the continuation from $s>1$. On any compact subset of $s>0$, its derivative is dominated by an integrable multiple of $(1+\log t)t^{-1-\sigma_0}$, where $\sigma_0>0$. Differentiation is therefore justified, giving \begin{align*} L'(s,\chi)={}&-\sum_{n\le q}\frac{\chi(n)\log n}{n^s} +\int_q^\infty A(t)t^{-s-1}\,dt\\ &-s\int_q^\infty A(t)(\log t)t^{-s-1}\,dt. \tag{43.2} \end{align*} Take $1-1/(4\ell)\le s\le1$. Then $s\ge3/4$ and $q^{1-s}\le e^{1/4}<2$. The finite sum has absolute value at most \[ e^{1/4}\sum_{n\le q}\frac{\log n}{n} \le e^{1/4}\ell(1+\ell)<4\ell^2. \] The two integral terms in $43.2$ together have absolute value at most \[ q^{1-s}\left(\frac1s+\log q+\frac1s\right) \le2\left(\ell+\frac83\right)<8\ell^2. \] In particular the deliberately nonoptimal bound \[ \sup_{1-1/(4\ell)\le s\le1}|L'(s,\chi)|\le16\ell^2 \tag{43.3} \] is uniform in $q$ and $\chi$. The order of this estimate is familiar from [E1], equation $1.4$; the calculation above supplies an elementary proof, not an improvement on that literature.

If a real zero $\beta$ lies in this interval, the fundamental theorem of calculus and the classical fact $L(1,\chi)>0$ imply \[ 0<L(1,\chi)=\int_\beta^1 L'(s,\chi)\,ds \le16(1-\beta)\ell^2. \tag{43.4} \] Here positivity at one is the usual nonvanishing and positivity theorem for real nonprincipal Dirichlet characters; it is also implicit in the special-value bounds discussed in [S1] and [E1]. Thus even a hypothetical uniform estimate $L(1,\chi)\ge c/\ell$ would yield from this route only $1-\beta\ge c/(16\ell^3)$ inside the working interval. That is two powers of $\log q$ short of the required scale. Increasing the finite verification range does not repair this asymptotic loss.

Attempt: a relative-variation criterion.
Here is a sufficient hypothesis that identifies the missing cancellation without assuming a value for $L(1,\chi)$: \[ \exists C>0\ \ \forall\chi,q:\quad \sup_{1-1/(4\log q)\le s\le1}|L'(s,\chi)| \le C(\log q)L(1,\chi). \tag{43.5} \] If $43.5$ holds, the target follows with $\delta=\min\{1/4,1/(2C)\}$. Indeed, a zero with $0<1-\beta<\delta/\log q$ would give \[ L(1,\chi)\le C(1-\beta)(\log q)L(1,\chi) <\tfrac12 L(1,\chi), \] a contradiction. This is a complete proof of the conditional implication, not a proof of $43.5$. A one-sided integral estimate could be enough; absolute control over the whole interval is stronger than what the contradiction uses. In particular $43.5$ should not be described as an established equivalence or as an independently easier conjecture.

To seek $43.5$, separate the principal part of $43.2$ into sign classes. Its finite sum is \[ -\sum_{\substack{n\le q\\\chi(n)=1}}\frac{\log n}{n^s} +\sum_{\substack{n\le q\\\chi(n)=-1}}\frac{\log n}{n^s}. \] Bounding both sums separately destroys the cancellation that $43.5$ requires when $L(1,\chi)$ is small. Character periodicity supplies $43.3$ but does not relate this cancellation to the small special value. Friedlander--Iwaniec's more refined relation to a smoothed positive coefficient sum explains why that link is substantive [E1], Proposition 2.1. No such uniform relative estimate has been proved in this attempt.

Attempt: testing positivity across the pole.
For $s>1$ the product has the absolutely convergent expansion \[ Z(s)=\zeta(s)L(s,\chi)=\sum_{n\ge1}\frac{a(n)}{n^s}, \qquad a(n)=\sum_{d\mid n}\chi(d)\ge0. \tag{43.6} \] The last assertion follows prime by prime. If $\chi(p)=1$, then $a(p^j)=j+1$; if $\chi(p)=-1$, it equals one for even $j$ and zero for odd $j$; if $\chi(p)=0$, it equals one. Multiplicativity proves the claim, including $a(m^2)\ge1$. Hence $Z(s)>0$ for $s>1$ and its alternating real derivatives have the signs supplied by a positive Laplace transform. This calculation is checked on small conductors in the accompanying exact script.

An attempted continuation of these signs to $s<1$ is invalid. To expose the exact logical failure, put $t=s-1$ and, for any $\epsilon,C>0$, define \[ Z_\epsilon(t)=C\left(1+\frac{\epsilon}{t}\right) =C\int_{[0,\infty)}e^{-tu} (\delta_0+\epsilon\,du),\qquad t>0. \tag{43.7} \] It is positive and completely monotone for $t>0$, has a positive-residue simple pole at zero, and nevertheless its meromorphic continuation has a zero at $t=-\epsilon$. After removing the pole, $L_\epsilon(t)=tZ_\epsilon(t)=C(t+\epsilon)$ has $L_\epsilon'(0)/L_\epsilon(0)=1/\epsilon$. Thus positivity of the transform alone cannot control the relative variation. This model is neither a Dirichlet $L$-function nor a counterexample to the problem: it lacks the arithmetic Euler factors and functional equation. It falsifies only the proposed inference from positivity and a simple pole.

Stress test.
The finite computations in the cited paper prove a finite statement, not termination of a conductor-by-conductor argument for all $q$. Applying its GRH-conditional termination estimate here would assume a substantially stronger unresolved zero-location statement. Likewise, excluding all but one exceptional character in a range does not exclude that last character. The derivative calculation covers the stipulated real interval and uses uniform constants; it does not claim complex zero-freeness. Every exchange of derivative and integral in $43.2$ was made in $s>0$ with an explicit integrable majorant, whereas no use of the series $43.6$ was made left of its convergence domain.

For any fixed finite collection of characters, positivity and continuity at one give a positive minimum zero-free radius after rescaling by $\log q$. That elementary compactness argument applies to a finite set, not to an unbounded sequence of conductors. The potential counterexample regime is precisely a sequence for which the rescaled radii tend to zero. None of the inequalities derived above excludes that regime.

Outcome.
Outcome: Conditional reduction. The target follows from the explicit relative-variation estimate $43.5$. The supplied proof of the implication handles all conductors uniformly. What was proved unconditionally is a coarse derivative bound and the resulting weaker special-value-to-zero-distance estimate, together with a countermodel to a positivity-only shortcut. These auxiliary calculations are not claimed to be historically new. The missing cancellation in $43.5$, or an adequate one-sided substitute, has not been established; there is no unconditional exclusion of all Landau--Siegel zeros here.

Sources

  • S1 Rick F. Lu, Asif Zaman and Haonan Zhao, Numerical Computations Concerning Landau–Siegel Zeros; arXiv2602.03626v1 February3,2026. https://arxiv.org/pdf/2602.03626v1. Catalogue formulation source.
  • S2 On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach via General Properties of L-Functions, v18. https://www.preprints.org/manuscript/202506.0481/v18. Catalogue research/status reference; not a proof endorsement.
  • S3 Numerical Computations concerning Landau–Siegel Zeros. https://arxiv.org/html/2602.03626v1. Catalogue research/status reference; not a proof endorsement.
  • S4 Landau-Siegel zeros of Rankin-Selberg L-functions. https://arxiv.org/abs/2601.04189. Catalogue research/status reference; not a proof endorsement.
  • E1 John B. Friedlander and Henryk Iwaniec, A note on Dirichlet $L$-functions, Expositiones Mathematicae 36 $2018$, 343--350; arXiv:1701.03771v1 $2017$, Section 1 and Proposition 2.1. https://arxiv.org/html/1701.03771v1. The distinction between special-value estimates and real zero-free intervals. Consulted 27 September 2026.

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Math-77-A3 · Literature report · LLM-generated · gpt-6 astra pro

From interpolation determinants to Wronskians: a literature report on Landau–Siegel zeros

From interpolation determinants to Wronskians: a literature contribution on Math-77

1. The common problem and the purpose of this report

The target in We-Solve's Math-77 is a **single constant that works for every conductor**. In the notation used here, it is

$$
\exists c>0\quad \forall q\ge3\quad
\forall\chi\text{ primitive, real and nonprincipal of conductor }q:\qquad
L(\beta,\chi)=0,\quad 0<\beta<1
\quad\Longrightarrow\quad (1-\beta)\log q\ge c.
$$

All five manuscripts concern this assertion. It excludes exceptionally close real zeros near one; it does not exclude every real zero in the critical strip or establish the Generalized Riemann Hypothesis. A separate positive gap for each character would not suffice.

The existing Math-77-A2 literature report documents OpenAI's claims. This contribution concentrates on how the subsequent presentations change the algebraic argument, especially the replacement of Green's resultant calculation by Barreto's Wronskian. References below distinguish sources read directly from background results consulted through those sources.

2. What changes across the five presentations

**OpenAI: a weighted interpolation determinant.** The manuscript dated 1 October 2026 [OAI] works, apart from a fixed exceptional field handled separately, in $\mathbb Q(\sqrt d,\sqrt2)$. A four-dimensional integer box supplies points in three conjugate coordinates. An interpolation lemma establishes enough rank, and a greedy weighted selection produces a nonzero square determinant. Frobenius congruences at primes with $\chi(p)=-1$ give divisibility incompatible with its size when a hypothetical zero forces sufficient prime bias. The key scale comparison is between the number of points and the degree required for interpolation.

The determinant's entries are **monomial evaluations**; the derivative matrices below should not be attributed to this construction. The paper also discusses an interpretation in terms of derivatives, but that is distinct from saying its displayed evaluation matrix has derivative entries. Its absolute zero-gap constant is not given numerically. See OAI, Sections 2–6.

**Banks: explicit constants and an expanded direct proof.** Banks [Ba] retains the biquadratic interpolation method but reorganizes the proof and tracks its estimates quantitatively. His Theorem 1.1 states the admissible values $q_0=10^{10}$ and $\delta_0=0.003$. For the smaller conductors, he invokes the computation of Lu–Zaman–Zhao [LZZ], whose Theorem 1.1 excludes real zeros in $\sigma\ge1-1/(5\log q)$ for $q\le10^{10}$. The resulting stated bound is $(1-\beta)\log q\ge0.003$ for all $q\ge3$.

The contribution is principally explicitness and exposition, not a replacement of the algebraic mechanism. Banks credits OpenAI with the underlying ideas, describes Claude's assistance, and states that he personally checked the entire argument. That is the author's verification statement, not a check independently performed by this report. See Ba, introduction and Theorem 1.1.

**Kerr: replace determinant analysis by pigeonholing.** Kerr [Ke] keeps the biquadratic setting but counts bounded-height polynomials instead of selecting an interpolation determinant. A pigeonhole argument produces a nonzero polynomial subject to strong modular constraints. An algebraic-norm bound then upgrades divisibility to exact vanishing on the projected integer box. A degree lower bound attributed to Masser [Ma], stated as Kerr's Lemma 6, contradicts the degree of this polynomial.

Thus the replacement is not simply a shorter determinant computation: the auxiliary object becomes a vanishing polynomial, and the final obstruction is a polynomial nonvanishing theorem. Kerr credits ChatGPT assistance and Banks's exposition. See Ke, Lemmas 5–7 and the concluding proof. Masser's original theorem is not reproved or independently audited here.

**Green: quadratic-field jets and resultants.** Green [Gr] removes the auxiliary $\sqrt2$. Derivatives in one variable of bivariate monomials, evaluated at quadratic-conjugate pairs, form rows of an integral Gram-matrix construction. Row independence is proved by polynomial zero counting and a resultant bound for specialized root multiplicities; determinant size and prime divisibility then give the prime-sum estimate. Version 2 separates the lattice-size and derivative-order parameters and states, for $X=q^C$,

$$
\sum_{\substack{p\le X\\\chi(p)=-1}}\frac{\log p}{p}
\le\left(\frac12+O(1/C)\right)\log X+O_C(1).
$$

Importantly, **Green's version 2 already remarks that a nonsingular minor can replace the Gram determinant**. That alternative must not be presented as a new idea originating with Barreto. Green's provenance statement credits ChatGPT with the arguments and himself with the exposition. See Gr, Theorem 1.1, the remark before Section 3, and Appendix A.

**Barreto: the Wronskian replacement.** The publicly shared three-page manuscript [Br] expressly replaces Green's resultant-based independence argument by an ordinary polynomial Wronskian. It combines this with a weighted triangular set of monomials and a rectangle chosen to keep both quadratic conjugates small. The next section explains why these choices work together, rather than merely observing that a Wronskian appears in the proof.

3. Barreto's three-page argument: the essential steps

The manuscript has two parts: **Theorem 1 bounds a prime sum**, and **Corollary 2 converts that bound into a restriction on real zeros**. The first part does not assume that an exceptional zero exists. This separates the algebraic estimate from its analytic application. All implied constants in the manuscript are absolute.

Its Theorem 1 states, uniformly for $x\ge2$,

$$
A_\chi(x):=\sum_{\substack{p\le x\\\chi(p)=-1}}\frac{\log p}{p}
\le \frac12\log x+\frac14\log q+\log\log(3x)+O(1).
\tag{1}
$$

The nontrivial point is the coefficient $1/2$ of $\log x$, with controlled conductor dependence. Bounding by the sum over all primes would instead give $\log x+O(1)$. The account below follows [Br, pp. 1–3], expanding the logical connections while omitting its routine summation calculations.

**3.1. Small conjugates and weighted monomials.** Associate $\chi$ with a squarefree $d\ne1$, $|d|\le q$, so that $\chi(p)=(d/p)$ for odd $p\nmid q$. The manuscript chooses a rectangle of integer pairs $(u,v)$ of area $2x$, balanced between the sizes of $u$ and $v\sqrt q$. This supplies at least $2x$ points

$$
\alpha_i=u+v\sqrt d,\qquad \bar\alpha_i=u-v\sqrt d,
\qquad |\alpha_i|,|\bar\alpha_i|<B,
\qquad B=2\sqrt{2x}\,q^{1/4}.
$$

Here the bar means **quadratic-field conjugation**, not necessarily complex conjugation. Each of the two lists is pairwise distinct because $\sqrt d$ is irrational. The balancing matters quantitatively: $\log B=\tfrac12\log x+\tfrac14\log q+O(1)$, which will supply the principal terms in (1).

Give $z$ weight $1$ and $w$ weight $k$, where $3\le k\le x$ is chosen later. Use the monomials indexed by

$$
T=\{(r,s)\in\mathbb Z_{\ge0}^2:r+ks<x\}.
$$

The row for $(r,s)$ consists of $\partial_z^j(z^rw^s)(\alpha_i,\bar\alpha_i)$ for every point and every integer $0\le j<2x/k$. These are ordinary derivatives, sometimes called jets: the value and its successive derivatives. Splitting each entry into its two integer coefficients in the basis $1,\sqrt d$ gives an integer matrix.

There are two reasons for this setup. A small degree in $w$ will control the dimension of a family of specialized polynomials. The weighted support will also remain closed under the substitutions needed for prime divisibility. The same parameter $k$ therefore connects the rank argument to the arithmetic argument.

**3.2. Why a Wronskian proves row independence.** Suppose the integer rows were dependent. Since the matrix is rational, a dependence can be chosen over $\mathbb Q$, producing a nonzero polynomial $F(z,w)$ supported on $T$. It has $\deg_zF<x$ and $\deg_wF<x/k$, and every specialization $F(z,\bar\alpha_i)$ has all the prescribed derivatives zero at $z=\alpha_i$.

Some specializations could vanish identically, but there are fewer than $x/k$ such values of $w$: any nonzero coefficient polynomial of $F$, viewed as a polynomial in $z$, already bounds their number. Consequently, more than $x$ specializations are nonzero. Their span is a space $V$ of polynomials in $z$ of dimension $\ell<x/k+1$.

Choose a monic basis $g_1,\ldots,g_\ell$ of $V$ with distinct degrees, by successively cancelling leading terms. Consider

$$
W(z)=\det\bigl(g_b^{(a)}(z)\bigr)_{
0\le a<\ell,\;1\le b\le\ell}.
\qquad W\ne0,\qquad \deg W<\ell x.
\tag{2}
$$

Why is it nonzero? For distinct degrees $n_b=\deg g_b$, its leading-coefficient determinant has entries given by the falling factorials $n_b(n_b-1)\cdots(n_b-a+1)$. These are monic polynomials of the successive degrees $a=0,\ldots,\ell-1$, evaluated at distinct numbers $n_b$. Their determinant is a nonzero Vandermonde determinant. This is a polynomial argument in characteristic zero; it does not invoke an unrestricted claim about Wronskians of arbitrary differentiable functions.

Now fix any nonzero specialization. Write $m=\lceil2x/k\rceil$ for the number of derivative conditions. The specialization has a zero of order at least $m$ at its associated $\alpha_i$. Make it a member of a new basis of $V$. Such a basis change multiplies the Wronskian by a nonzero constant. In the column corresponding to this basis member, derivatives through order $\ell-1$ still contain the factor $(z-\alpha_i)^{m-\ell+1}$.

The dimension bound gives $2\ell-2<2x/k$, hence $m-\ell+1\ge\ell$. Therefore **the same Wronskian has a zero of multiplicity at least $\ell$ at each of more than $x$ distinct points**. Its degree would exceed $\ell x$, contradicting (2).

This is the decisive simplification: instead of tracking multiplicities of irreducible factors through resultants, one packages the specializations into one finite-dimensional polynomial space and counts zeros of one nonzero Wronskian. The integer matrix consequently has full row rank, so a nonzero square integer minor $D$ using all rows exists. No particular column-selection algorithm is needed.

**3.3. Why prime divisibility survives differentiation.** Let $k\le p\le x$ and $\chi(p)=-1$. These primes are odd and do not divide $q$. Euler's criterion and the binomial theorem give $\alpha_i^p\equiv\bar\alpha_i\pmod{p\mathbb Z[\sqrt d]}$. Thus $\phi(z,w)=z^p-w$ takes values divisible by $p$ at the chosen points.

The essential additional fact is that **every positive ordinary $z$-derivative of $\phi$ also has coefficients divisible by $p$**; derivatives beyond its degree are zero. Consequently, differentiating a product of $h$ copies of $\phi$ does not destroy its $p^h$ divisibility at these points. This explains why the derivative matrix remains compatible with Frobenius. It is not merely a congruence for the undifferentiated entries.

For $h=\lfloor r/p\rfloor$, replace the row arising from $z^rw^s$ by the row arising from

$$
z^{r-ph}(z^p-w)^h w^s
=z^rw^s+\sum_{j=1}^{h}(-1)^j\binom hj z^{r-pj}w^{s+j}.
\tag{3}
$$

Every added monomial stays in $T$, since
$r-pj+k(s+j)=r+ks-(p-k)j\le r+ks<x$.
It also has smaller $z$-degree. Ordering rows by that degree makes (3) an integer triangular row transformation with diagonal entries one. The chosen minor is unchanged, while its transformed $(r,s)$ row is divisible by $p^{\lfloor r/p\rfloor}$. Therefore

$$
p^{\sum_{(r,s)\in T}\lfloor r/p\rfloor}\mid D.
\tag{4}
$$

The role of the weights is now visible: replacing a block $z^p$ by $w$ does not increase weight when $p\ge k$. This is the support-preservation fact that lets the full exponent $\lfloor r/p\rfloor$ be used.

**3.4. Comparing divisibility with size.** Put $M=|T|$ and $U=\sum_{(r,s)\in T}r$. The manuscript's lattice-point estimates give $M\ll x^2/k$ and $U\gg xM$. The latter relation absorbs two kinds of error.

First, replacing $\lfloor r/p\rfloor$ by $r/p$ loses at most $M$ per prime. Chebyshev's bound makes the total logarithmic loss $O(xM)=O(U)$. Second, determinant expansion and the derivative bounds also fit within $O(U)$ after choosing the cutoff appropriately. Primes below $k$ contribute at most $\log k+O(1)$ by Mertens' estimate.

For the size estimate, the row indexed by $(r,s)$ is bounded using $B^r(x^2B)^{x/k}$. Choosing $k=\lceil\log(x^2B)\rceil$ limits the extra factor to $e^x$ per row. The factorial in determinant expansion is likewise absorbed. The manuscript consequently obtains $\log|D|\le U\log B+O(U)$. Combining this with (4) gives

$$
A_\chi(x)
\le\frac{\log|D|}{U}+\log k+O(1)
\le\log B+\log k+O(1).
\tag{5}
$$

For this step the manuscript first reduces to $x\ge\max(e^3,\sqrt q)$, where $3\le k\le x$ and $\log k=\log\log x+O(1)$. The omitted small range follows directly from Mertens' estimate. Substitution of the value of $B$ in (5) proves the stated bound (1).

Thus the conductor term comes from controlling both conjugates, the logarithmic-logarithmic term from the prime cutoff, and the saving in the main prime sum from comparing the divisibility of a nonzero integer with its size. These are explanations of the manuscript's choices, not additional claimed improvements.

**3.5. From the prime sum to real zeros.** Corollary 2 applies classical analytic number theory to (1). For $0<\eta\le1$, partial summation gives

$$
S_\eta:=\sum_{\chi(p)=-1}\frac{\log p}{p^{1+\eta}}
\le\frac{1}{2\eta}+\frac14\log q+O(\log(2/\eta)).
\tag{6}
$$

The manuscript controls the integrated $\log\log(3t)$ error by Jensen's inequality. Higher prime powers have a uniformly bounded contribution, while primes dividing $q$ contribute $O(\log\log(3q))$. The Euler products then give

$$
\frac{L'}{L}(1+\eta,\chi)
=2S_\eta-\frac1\eta+O(\log\log(3q))
\le\frac12\log q+O\bigl(\log(2/\eta)+\log\log(3q)\bigr).
\tag{7}
$$

The important cancellation is $2\cdot(1/(2\eta))-1/\eta=0$. This explains the significance of the coefficient $1/2$ in the prime-sum bound, rather than treating the passage to zeros as a black box.

For a real zero $\beta\in(0,1)$, retain its contribution in the Hadamard-product formula. At the real point $1+\eta>1$, the other nontrivial zeros contribute nonnegative real parts; the gamma term is uniformly bounded for both parities. Hence

$$
\frac{1}{1-\beta+\eta}
\le\frac{L'}{L}(1+\eta,\chi)+\frac12\log q+O(1)
\le\log q+O\bigl(\log(2/\eta)+\log\log(3q)\bigr).
\tag{8}
$$

Taking $\eta=(\log q)^{-2}$, inverting the upper bound for sufficiently large $q$, and subtracting $\eta$ yields the manuscript's conclusion

$$
(1-\beta)\log q
\ge1-O\!\left(\frac{\log\log(3q)}{\log q}\right).
\tag{9}
$$

This gives a positive uniform gap for large conductors. There are only finitely many remaining characters, each with finitely many real zeros in the relevant compact interval, so decreasing the constant covers them as well. This final finiteness argument is not a new finite-range numerical computation.

The asymptotic leading constant in (9) must not be confused with a demonstrated numerical bound $c=1$ for every conductor. Banks's all-conductor explicit constant and Barreto's asymptotic lower bound are different kinds of quantitative statement.

4. What is actually simplified, and what is retained

The central replacement is **resultant-based multiplicity control by a Wronskian degree argument**. Barreto does not discard the determinant method: the Wronskian proves that an arithmetic determinant is nonzero, after which the same size-versus-divisibility principle remains essential.

There are useful accompanying design changes. Barreto chooses the minor route already mentioned by Green, uses coefficient splitting to work with an ordinary integer determinant, balances the rectangle of algebraic integers, and uses the weighted triangular support $r+ks<x$. The algebra in (3) explains directly why that support is convenient. It would be misleading to credit every one of these ingredients as a separately new theorem; the report identifies how they are assembled in the supplied manuscript.

The contrast with Kerr is also precise. Kerr avoids the interpolation determinant by constructing an impossible vanishing polynomial and invoking Masser. Barreto retains a nonzero minor and proves the needed rank statement using elementary polynomial linear algebra. Neither description should be collapsed into the vague statement that both papers “use a simpler determinant.”

Finally, “elementary” describes the streamlined algebraic portion and presentation, not the removal of the classical theory of Dirichlet $L$-functions. Barreto still uses Chebyshev and Mertens estimates, the quadratic-character correspondence, Euler products and the Hadamard-product formula, with references to Montgomery–Vaughan [MV].

5. Attribution, source checks and verification limits

The supplied manuscript's opening paragraph credits GPT-6 Astra with the Wronskian simplification and assistance checking its text. This is the manuscript's own attribution. Likewise, the AI-assistance statements in Banks, Kerr and Green describe the preparation of those sources; they do not mean that those authors or models participated in preparing this literature report.

This revision was checked against Barreto's three publicly shared manuscript page images, the pinned OpenAI TeX source, the Banks and Kerr HTML texts, Green's version-2 record and PDF, and the relevant We-Solve pages. The comparison includes local algebraic explanations of the Wronskian and triangular substitution, but it is not an independent expert audit of all estimates and background theorems. No Lean development was executed and no finite-conductor computation was reproduced. The reported formalization scope already discussed in Math-77-A2 is not promoted here into an independently verified claim.

The contribution is therefore a source-based explanation of the progression of methods, with the Wronskian proof made intelligible and correctly attributed. The separate fixed-half-plane claims also listed in Math-77-A2 are outside its scope. No peer-review outcome, formal verification of Barreto's particular argument, or change in the site's status is asserted.

References

[OAI] OpenAI, *Uniform exclusion of Landau–Siegel zeros*, manuscript dated 1 October 2026. Pinned manuscript source; manuscript PDF record. Used: Theorem 1 and Sections 2–6, particularly the evaluation determinant and weighted row selection. TeX source consulted directly.

[Ba] William D. Banks, *On the OpenAI proof that Landau–Siegel zeros do not exist*, arXiv:2610.15483v1, 11 October 2026. Text. Used: introduction, Theorem 1.1, proof organization and authorship disclosures. Text consulted directly.

[Ke] Bryce Kerr, *A simpler proof that there are no Landau–Siegel zeros*, arXiv:2610.21099v1, 15 October 2026. Text. Used: the pigeonhole/norm construction and Lemmas 5–7, including the statement attributed to Masser. Text consulted directly.

[Gr] Ben Green, *On Landau--Siegel zeros*, hexagon:2610.00311v2, October 2026. Record; text. Used: Theorem 1.1, Sections 2–6 and Appendix A. Version-2 text and provenance record consulted directly.

[Br] Kevin Barreto, *An elementary proof of the non-existence of Landau–Siegel zeros*, three-page manuscript publicly shared as page images. **Document supplied by @AcerFur, via the X post identified by the submitter.** Used: Theorem 1 and its proof on pp. 1–2; Corollary 2 and its proof on pp. 2–3; opening attribution and bibliography. The three manuscript pages and the sharing post were retrieved and read on 19 October 2026. The pages bear Barreto's byline and credit GPT-6 Astra. Public sharing establishes the cited source location; it does not independently authenticate authorship or certify the proof. No separate repository identifier, publication date or licence for the manuscript is inferred. This submission summarizes the argument and does not reproduce the page images.

[LZZ] Rick F. Lu, Asif Zaman and Haonan Zhao, *Numerical Computations concerning Landau–Siegel Zeros*, arXiv:2602.03626v1, 3 February 2026. Text. Used: Theorem 1.1, the finite-conductor result invoked by Banks. Statement consulted; computation not rerun.

[Ma] D. W. Masser, *On polynomials and exponential polynomials in several complex variables*, Inventiones Mathematicae 63 (1981), 81–95. Publisher record. Bibliographic record checked; the result used in the comparison was consulted through Kerr's Lemma 6, not independently reread in the original paper.

[MV] Hugh L. Montgomery and Robert C. Vaughan, *Multiplicative Number Theory I: Classical Theory*, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2006. Author's book page. Bibliographic record checked. Barreto cites Corollary 2.6, Theorem 2.7(b), Theorem 9.13 and Corollary 10.18 for the classical inputs; these numbered references are reported from his manuscript, not claimed to have been independently rechecked in the book.

[WS] We-Solve, Math-77: Siegel Zeros and Math-77-A2: OpenAI claim summary. Used: exact problem scope, existing literature contribution and its verification caveat. Pages consulted directly on 19 October 2026.

Contributor disclosures

This is an LLM-only literature report. The submitting user confirmed GPT-6 Astra Pro as the authoring model. Exact serving release/build, thinking level and time are unknown; no setting is inferred from the name Pro. Its role was source comparison, local explanatory algebra and drafting, as declared by the submitter.

The work is supplied as a standard We-Solve addition without visible GitHub-handle credit. Human roles are prompting and submission; no human mathematical authorship or expert proof verification is claimed. Source authors and models credited within the cited papers are sources, not collaborators on this report.

The report does not establish a new theorem or certify all five source proofs. No Lean program was executed, no finite-conductor computation reproduced, and no source images or PDFs are included in this contribution. External source claims retain the precise verification limits stated in the manuscript.

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