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\hypersetup{pdfauthor={Roelof van Dijk},
  pdftitle={A Frobenius determinant proof of the non-existence of Landau--Siegel zeros}}
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\begin{document}
\begin{center}
{\Large\bfseries A Frobenius determinant proof}\\[3pt]
{\Large\bfseries of the non-existence of Landau--Siegel zeros}\\[9pt]
Roelof van Dijk
\end{center}

\begin{abstract}
We give a variant of the OpenAI proof~\cite{OpenAI} of the non-existence
of Landau--Siegel zeros, following the simplifications of Kerr~\cite{Kerr},
Green~\cite{Green} and Barreto~\cite{Barreto}.
The Frobenius congruence proves both divisibility and nonvanishing of the
determinant. At a large inert prime supplied by the hypothetical zero,
it reduces nonvanishing to a polynomial root count, replacing resultants
and Wronskians.
The argument was discovered in discussions with GPT-6 Astra and Opus 5.5
and the new arguments should be attributed to these models.
\end{abstract}

\begin{theorem}[OpenAI~\cite{OpenAI}]
There is an absolute $c>0$ such that every real zero $\beta\in(0,1)$
of a primitive nonprincipal real Dirichlet $L$-function of conductor $q\ge3$
satisfies $(1-\beta)\log q\ge c$.
\end{theorem}

\begin{proof}
Suppose the theorem is false. For every integer $k\ge4$, choose such
a character and zero with $(1-\beta)\log q\le k^{-2}$.
Choose a squarefree $d\ne1$ with $|d|\le q$ such that
$\chi(p)=(d/p)$ for odd primes $p$.
We call a prime inert when $\chi(p)=-1$.
All these quantities may depend on $k$; all limits are as $k\to\infty$.

Put $\delta=1-\beta$, $Y=q^k$ and $t=1/\log Y$.
Then $\log q=o(\log Y)$ and $\delta\log Y\le1/k$.
The Euler products and the Hadamard formula~\cite[Cor.~10.18]{MV} give
\[
 \frac2e\sum_{\substack{p\le Y\\\chi(p)=1}}\frac{\log p}{p}
 \le-\frac{\zeta'}\zeta(1+t)-\frac{L'}L(1+t,\chi)
 \le\frac1t-\frac1{t+\delta}+\frac12\log q+O(1)
 =o(\log Y).
\]
Here $t^{-1}-(t+\delta)^{-1}\le\delta t^{-2}=o(\log Y)$.
Mertens' estimate and $\sum_{p\mid q}(\log p)/p\le\log q$ now imply
\[
 \sum_{\substack{p\le Y\\\chi(p)=-1}}\frac{\log p}{p}\sim\log Y.
\]
For sufficiently large $k$, let $p_0$ be the largest inert prime at most $Y$.
The preceding sum is unchanged when $Y$ is replaced by $p_0$,
so Mertens' estimate gives $\log p_0\sim\log Y$.
Set
\[
 m=\lfloor p_0/k^2\rfloor,\qquad R=k^2m,\qquad
 S=km,\qquad T=S-m,\qquad N=RS.
\]
Since $\log p_0\sim k\log q$ and $q\ge3$, we have $k^2=o(p_0)$, hence
$R\sim p_0$ and $\log R\sim\log Y$. Since $p_0/S\sim k$, Mertens' estimate
shows that omitting primes outside $[k^2,S]$ costs $O(\log k)$. Thus
\[
 \sum_{\substack{k^2\le p\le S\\\chi(p)=-1}}\frac{\log p}{p}\sim\log R.
\]

For every odd inert prime $p$, the binomial theorem, Fermat's little theorem
and Euler's criterion give
\begin{equation}\label{frobenius}
 (u+v\sqrt d)^p
 \equiv u+v\,d^{(p-1)/2}\sqrt d
 \equiv u-v\sqrt d\pmod{p\Z[\sqrt d]}
 \qquad(u,v\in\Z).
\end{equation}
Let $M=\lceil\sqrt{p_0}\rceil$, and form the matrix
\[
 \mathcal A_{(r,s),(u,v,j)}
 =(\partial_z^j z^rw^s)(u+v\sqrt d,u-v\sqrt d),
\]
with rows indexed by $0\le r<R$, $0\le s<S$, and columns by
$0\le u,v<M$, $0\le j<S$, all indices integral.
We claim that $\mathcal A$ has row rank $N$ modulo $p_0$.
In $K=\Z[\sqrt d]/p_0\Z[\sqrt d]\cong\mathbb F_{p_0^2}$,
the $M^2\ge p_0$ residues $u+v\sqrt d$ are distinct because
$M\le p_0$ and $1,\sqrt d$ form an $\mathbb F_{p_0}$-basis.
\newpage
By~\eqref{frobenius}, a nontrivial row dependence would give a polynomial
\[
 G(z)=\sum_{r<R,\,s<S}c_{rs}z^{r+p_0s}\in K[z]
\]
whose derivatives of orders $0,\ldots,S-1$ vanish at all these residues:
in characteristic $p_0$, $(z^{p_0})'=0$.
Since $R\le p_0$, the exponents $r+p_0s$ are distinct, so $G\ne0$ and
$\deg G\le R-1+p_0(S-1)<p_0S$.
Since $S<p_0$, the factorials $j!$ for $j<S$ are invertible in $K$.
Thus $G$ has at least $p_0$ distinct roots of multiplicity at least $S$,
contradicting its degree bound.
Choose an $N\times N$ minor $\Delta\in\Z[\sqrt d]$ nonzero modulo $p_0$;
it uses every row. Set
$D=|\Nm_{\Q(\sqrt d)/\Q}\Delta|\in\Z_{>0}$.

For each odd inert prime $k^2\le p\le S$, replace the rows with $s<T$ by
\[
 z^rw^s\longmapsto z^{r-ph}(z^p-w)^h w^s,
 \qquad h=\lfloor r/p\rfloor.
\]
Here $h<m$, so $s+h<S$, and all added monomials have smaller $z$-degree.
Ordered by increasing $r$, this is a unitriangular row transformation
and therefore preserves $\Delta$.
By~\eqref{frobenius}, $z^p-w\equiv0\pmod p$ at every grid point;
its positive $z$-derivatives have coefficients divisible by $p$.
By Leibniz's rule, every entry of the new row is divisible by $p^h$.
Taking norms gives
\[
 p^{\,2T\sum_{r<R}\lfloor r/p\rfloor}\mid D.
\]
The estimate $\sum_{r<R}\lfloor r/p\rfloor=R^2/(2p)+O(R)$ gives, uniformly
for $p\le S=R/k$,
\[
 \sum_{r<R}\lfloor r/p\rfloor
 \ge\bigl(1-O(k^{-1})\bigr)\frac{R^2}{2p}.
\]
Summing over these primes and using $T/S\to1$ gives
\begin{equation}\label{lower}
 \log D\ge
 \bigl(1-O(k^{-1})\bigr)TR^2
 \sum_{\substack{k^2\le p\le S\\\chi(p)=-1}}\frac{\log p}{p}
 =\bigl(1-o(1)\bigr)NR\log R.
\end{equation}

Under either embedding, the grid coordinates have modulus at most
$B=2M\sqrt q$, and
\[
 \log B=\tfrac12\log p_0+\tfrac12\log q+O(1)
       =\bigl(\tfrac12+o(1)\bigr)\log R.
\]
Since $j<S$, each entry in row $(r,s)$ has modulus at most $R^SB^{r+s}$.
Using $\sum_{r<R,\,s<S}(r+s)=N(R+S-2)/2$ and expanding the two
conjugate determinants gives
\begin{align*}
 \log D
 &\le2\log(N!)+2SN\log R+N(R+S)\log B\\
 &\le\bigl(\tfrac12+o(1)\bigr)NR\log R,
\end{align*}
since $S/R=1/k\to0$ and $\log(N!)=o(NR\log R)$.
This contradicts~\eqref{lower}.
\end{proof}

{\small
\begin{thebibliography}{5}
\setlength{\itemsep}{0pt}
\bibitem{OpenAI} OpenAI, \emph{Uniform exclusion of Landau--Siegel zeros},
OpenAI Math Release, 1 October 2026.
\url{https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf}.
\bibitem{Kerr} B.~Kerr, \emph{A simpler proof that there are no Landau--Siegel
zeros}, arXiv:2610.21099v1, 2026.
\url{https://arxiv.org/abs/2610.21099v1}.
\bibitem{Green} B.~Green, \emph{On Landau--Siegel zeros},
hexagon:2610.00311v2, 2026.
\url{https://hexagonmath.org/pdf/2610.00311v2}.
\bibitem{Barreto} K.~Barreto, \emph{An elementary proof of the non-existence
of Landau--Siegel zeros}, manuscript.
\url{https://x.com/AcerFur/status/2112239827077026128}.
\bibitem{MV} H.~L. Montgomery and R.~C. Vaughan,
\emph{Multiplicative Number Theory I: Classical Theory},
Cambridge University Press, 2006.
\end{thebibliography}
}
\end{document}