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Statement

For a primitive Dirichlet character $\chi$ of conductor $q$, $L(s,\chi)$ is the series $\sum_{n\ge1}\chi(n)n^{-s}$ on $\operatorname{Re}s>1$ and its analytic continuation elsewhere (Section 1). Theorem 1. There is an absolute constant $c>0$ with the following property: if $\chi$ is a primitive nonprincipal real Dirichlet character of conductor $q\ge3$ and $L(\beta,\chi)=0$ with $\beta\in(0,1)$ real, then

$$(1-\beta)\log q\ge c.$$

The constant $c$ is not made explicit: the proof argues by contradiction along a sequence of conductors and yields no value. The manuscript calls the statement the "logarithmic formulation" of the Landau--Siegel zero problem. It contrasts it with Page's theorem, whose window $(1-c_2/\log Q,1)$ depends on a common bound $Q$ for the conductors rather than on each character's own conductor, and with Siegel's ineffective bound $L(1,\chi)\gg_\varepsilon q^{-\varepsilon}$, which leaves open a sequence of real zeros with $(1-\beta)\log q\to0$. The theorem excludes real zeros in $(1-c/\log q,1)$ and says nothing about real zeros elsewhere in $(0,1)$, nor about complex zeros.

Source. OpenAI, Uniform exclusion of Landau--Siegel zeros, OpenAI Math Release preprint of 1 October 2026, release folder preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026; TeX source paper.tex, label thm:main with display eq:main, in Section 1 (PDF p. 1); the proof occupies Sections 2--6 (PDF pp. 2--8). Read on 2026-10-07 in the release's TeX source. The card records the release's attestations and the Lean comparator statement the release lists for this theorem.

Read depth. Claims checked: the statement, the definitions it uses, and the statements of Lemma 2, Lemma 3, Corollary 4, Lemma 6 and Lemma 7 were read clause by clause in the TeX source. The proof was read for its structure (below) and no step was checked. Nothing here is independently reviewed; the Lean comparator statement the release lists was read statically and not built or audited here.

Proof pointer

Sections 2--6 (pp. 2--8). Write $\ell=\log q$ and $\delta=(1-\beta)\ell$. Lemma 2 (Section 2) is the analytic input: the logarithmic-derivative identity from the Hadamard product and functional equation, with only the zero $\beta$ kept and $s=1+1/\log X$, shows that the primes $p\le X$ with $\chi(p)=1$ carry logarithmic mass $\ll\ell+\delta(\log X)^2/\ell$, so by Mertens' estimate the primes in $(H,X]$ with $p\nmid2q$ and $\chi(p)=-1$ carry at least $\log X-C\ell-C\delta(\log X)^2/\ell-C_H$. Sections 3 and 4 build the algebraic object: for the biquadratic field $K=\mathbb Q(\sqrt d,\sqrt2)$ attached to $\chi$ (the case $d=2$ is excluded, which the limit $q\to\infty$ permits), the $N^4$ numbers $\theta_n=n_1+n_2a+n_3b+n_4ab$ and their conjugates $\sigma(\theta_n)$, $\sigma\tau(\theta_n)$ furnish monomial rows $R_\alpha$ indexed by $\alpha\in\mathbb Z_{\ge0}^3$; Lemma 3, an interpolation estimate with separate degree bounds, and its Corollary 4 show that rows with $\alpha_1\le32H^{2/3}U$ and $\alpha_2,\alpha_3\le32H^{-1/3}U$, $U=N^{4/3}$, already span, so a greedy selection by the weight $\alpha_1+H\alpha_2+H\alpha_3$ yields a nonzero $M\times M$ determinant $\Delta\in\mathbb Z[a,b]$, $M=N^4$, whose exponent sums satisfy $S_2/S_1\le(C_0/c_0)/H$ (Lemma 6). Section 5 bounds the integer $\mathrm N(\Delta)$ two ways. Hadamard's inequality with $|\nu(\theta_n)|\le8N\sqrt q$ at every embedding $\nu$ gives $\tfrac14\log|\mathrm N(\Delta)|\le\tfrac M2\log M+(S_1+S_2)(\log N+\tfrac12\ell+\log8)$. For an admissible prime ($p>H$, $p\nmid2q$, $\chi(p)=-1$) the Frobenius relation $\theta^p\equiv g_p(\theta)\pmod p$, with $g_p\in\{\sigma,\sigma\tau\}$, lets each row be replaced, modulo lower-weight rows already in the span, by one divisible by $p^{\lfloor\alpha_1/p\rfloor}$ (Lemma 7), so $\tfrac14\log|\mathrm N(\Delta)|\ge S_1\sum_{p\le U\text{ admissible}}(\log p)/p-M\sum_{p\le U}\log p$, and Lemma 2 with Chebyshev's bound turns this into $S_1(\log U-C\ell-C\delta(\log U)^2/\ell-C_H)-CMU$. Section 6 compares the two: the divisibility side has leading term $S_1\log U$ and the size side $(S_1+S_2)\log N=(S_1+S_2)\tfrac34\log U$. Along a hypothetical sequence with $q\to\infty$ and $\delta\to0$, fixing $H$ makes $S_2/S_1\le1/12$ and the size term at most $13/16$ of the divisibility term, taking $N=\lceil q^\gamma\rceil$ with $\gamma$ large makes the $\ell/\log U$ contribution below $1/16$, and the $\delta$ term and the lower-order terms vanish, leaving $1\le7/8$. The hypothesis "real" enters through $\chi(p)=(d/p)$ and the Frobenius relation, and in Lemma 2, whose series $\sum_n\Lambda(n)(1+\chi(n))n^{-s}$ is nonnegative only for real $\chi$; "primitive nonprincipal" through the fundamental discriminant and $d\ne1$; $q\ge3$ is the range in which primitive nonprincipal real characters exist, the TeX does not single out where it enters, and Lemma 2 divides by $\ell=\log q$.

Dependencies

External inputs taken at statement level: the Hadamard product and functional equation of the completed Dirichlet $L$-function and the resulting logarithmic-derivative identity (Davenport, Multiplicative number theory, Sections 12 and 14); the bound $-\zeta'/\zeta(s)=1/(s-1)+O(1)$ for $1<s\le2$; Mertens' estimate $\sum_{p\le X}(\log p)/p=\log X+O(1)$; Chebyshev's bound $\sum_{p\le U}\log p\ll U$; the correspondence between primitive real characters and fundamental discriminants with $\chi(p)=(d/p)$ for odd $p\nmid q$ (Davenport); Euler's criterion; Hadamard's determinant inequality; the nonvanishing $L(1,\chi)\ne0$ for nonprincipal $\chi$; and the elementary structure of the biquadratic field $\mathbb Q(\sqrt d,\sqrt2)$ for squarefree $d\ne1,2$. The manuscript supplies its own proofs of Lemma 3 and Corollary 4; neither was checked here. The cited transcendence literature (Philippon, Fischler, Laurent, Bost) is context, not an input; Siegel's theorem and Page's theorem are cited for comparison only. None was checked here.

Bears on

The manuscript names no Erdős problem. The rows below state the relation to pages whose linked sources rest on a Siegel-zero hypothesis or input; the card carries the rows for those sources and for the pages where the result does not apply. Every relation is to an unverified claim, and no page's status rests on it.

• Problem 1204: the theorem claims to refute the hypothesis "infinitely many Siegel zeros" under which Granville's card, whose Bears-on row targets the page, derives from its Corollary 3 that $A(k_j)/(k_j\log k_j)\to1/2$ along a sequence; it proves nothing about $A(k)$ or $B(k)$. Unverified here; the page's status is unchanged.
• Problem 855: the same hypothesis underlies the conditional interval constructions on Granville's card, which bears on the page; nothing about $\pi(x+y)\le\pi(x)+\pi(y)$ follows. Unverified here; the page's status rests on its own evidence.