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\title{Broadbent et al.'s global theta coefficient improves from 0.024334 to 0.0143406585}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
A global upper bound for the Chebyshev function $\theta(x)=\sum_{p\leq x}\log p$ is obtained by splicing a verified small-$x$ sign result, 193 published numerical step envelopes, and a decreasing analytic tail. The exact maximum of the normalized rounded step envelopes is $0.0143406585$.
\end{abstract}

\[
\theta(x)<x\left(1+\frac{0.0143406585}{\log^3x}\right)\qquad(x>1)
\]

\section{Statement and normalization}

The claim controls only the upper signed error of $\theta(x)$. Writing the relative error as $E_\theta(x)$ shows that the coefficient is a uniform upper bound for $E_\theta(x)\log^3x$.

The resulting coefficient replaces $0.024334$ in the same global inequality and on the same range.

\[
E_\theta(x)=\frac{\theta(x)-x}{x}
\]

\[
E_\theta(x)\log^3x<0.0143406585\qquad(x>1)
\]
\section{Small values and overlap}

A theorem of Buethe gives $\theta(x)<x$ throughout $1\leq x\leq10^{19}$. This is stronger than the target upper bound because the added $\log^{-3}x$ term is positive.

The numerical table regime begins at $\log x=37$, while $\log(10^{19})$ lies between $43$ and $44$. The two inputs therefore overlap rather than leaving a gap.

\[
x-\theta(x)>0.05\sqrt{x}\qquad(1\leq x\leq10^{19})
\]

\[
43<19\log10<44
\]
\section{Finite step envelopes}

Each Fiori--Kadiri--Swidinsky row $(L_i,\epsilon_i)$ gives $E_\theta(x)\leq\epsilon_i$ for all $x\geq e^{L_i}$. On an interval between table knots, every earlier row remains available, so the best inherited value is the running minimum $\epsilon_i^*$.

Since $L^3$ increases across the interval, its limiting right-end product controls that interval. Exact rational comparison of all 193 rows finds one maximal interval.

\[
\epsilon_i^*=\min_{j\leq i}\epsilon_j
\]

\[
E_\theta(x)L^3\leq\epsilon_i^*L^3<\epsilon_i^*L_{i+1}^3
\]

\[
(1.5485\times10^{-12})\,2100^3=0.0143406585
\]

\begin{enumerate}[leftmargin=*]
\item Form the running minimum of the admissible table rows.
\item Evaluate each interval's right-end majorant exactly.
\item Check each knot using the newly available row.
\end{enumerate}
\section{Typo repair and strictness}

The first two displayed theta entries in Table 2 have an exponent inconsistency. Proposition 17 applied to the same row yields approximately $1.9536449\times10^{-8}$, so the outward rounded admissible value is $1.9537\times10^{-8}$ rather than the printed $10^{-9}$ exponent.

This repaired row does not determine the maximum. The maximum comes from $2075\leq L<2100$ and is an unattained right-end supremum. At $L=2100$, a stronger row becomes available and gives $0.0122671206$, so every actual knot value is strictly below the coefficient.

\[
\epsilon_{\theta,\mathrm{num}}(10^{19})=1.953644927822\ldots\times10^{-8}<1.9537\times10^{-8}
\]

\[
(1.3246\times10^{-12})\,2100^3=0.0122671206<0.0143406585
\]
\section{Analytic tail}

For $L=\log x\geq3100$, the Fiori--Kadiri--Swidinsky analytic envelope replaces the tables. After multiplying by $L^3$, logarithmic differentiation shows the normalized tail decreases once $L>81R/4$.

With $R=5.5666305$, this decrease begins far before the handoff. Directed evaluation at $L=3100$ gives a normalized tail below $0.000151$, well below the finite-envelope coefficient.

\[
E_\theta(x)\leq121.0961\left(\frac LR\right)^{3/2}\exp\!\left(-2\sqrt{\frac LR}\right)
\]

\[
\frac{h'(L)}{h(L)}=\frac9{2L}-\frac1{\sqrt{RL}}<0\qquad\left(L>\frac{81R}{4}\right)
\]

\[
h(3100)<0.000151<0.0143406585
\]

\section{Pinned certificate}
The pinned certificate checks the repaired FKS source row, all 193 exact rational step envelopes, the unique interval supremum at [2075,2100), strict knot values, overlap with the small-x range, and the decreasing Corollary 14 tail.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0047/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0047/verify.py}
\item \path{canon/witnesses/C-0047/PIN.md}
\end{itemize}

\section{Scope}
The coefficient is exact for the stated splice of published rounded-up upper step envelopes and controls the upper signed error.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0047-broadbent-global-theta-log3-coefficient-0-0143406585.md}
\item Pinned certificate note: \path{canon/witnesses/C-0047/PIN.md}
\item Certificate source: \path{experiments/fks_global_theta_log3_coefficient/verify.py}
\end{itemize}

\end{document}
