Broadbent et al.'s global theta coefficient improves from 0.024334 to 0.0143406585

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.

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.

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.

1. Form the running minimum of the admissible table rows.
2. Evaluate each interval's right-end majorant exactly.
3. Check each knot using the newly available row.

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.

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.