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\hypersetup{pdftitle={Linear independence of even Dirichlet beta values},
 pdfauthor={},pdfsubject={Working mathematical draft; independently cross-checked by another model},
 pdfcreator={LaTeX}}
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\title{Linear independence of even Dirichlet beta values}
\author{\normalsize Research note prepared with Codex}
\date{\small Working draft\\2 November 2026}
\begin{document}
\maketitle
\begin{abstract}
Let $\beta(s)=1-3^{-s}+5^{-s}-\cdots$ denote the Dirichlet beta function.
For all sufficiently large even integers $s$, we prove
\[
 \dim_{\Q}\Span_{\Q}\{\beta(2),\beta(4),\ldots,\beta(s)\}
 \geq 0.21\sqrt{\frac{s}{\log s}}.
\]
\end{abstract}
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