Erdős 699: sharper uniform prime bounds for numerical Padé sources

P699 research · Unequal Padé blocks

---

The numerical coefficient-gcd certificates improve substantially when their prime intervals use sharper estimates for $\theta(x)$. These estimates do not change the Padé construction or its target cofactor inequalities. They reduce the computed starting index at which those inequalities apply.

The uniform helper returns rational bounds for every $m\ge m_0$. Its public gcd_exponent_lower(A,B,C,terms=...,m0=...) interface matches the existing off-diagonal helper. theta_errors(numerator,denominator) supplies uniform relative errors for every real $x$ above a rational starting point; theta_pointwise supplies rational bounds at that point for finite bootstrap calculations.

A square-root bound through $10^{19}$

Broadbent et al., Corollary 9.1 with $c=C=.94$ and $c_0=1.03883$ gives

$$\theta(x)\ge x-1.94x^{1/2}-1.03883x^{1/3} -.94x^{1/4}-1.03883x^{1/5} \qquad(10^4\le x\le10^{19}).$$

For $x\ge10^{11}$,

$$x^{1/6}>64,\qquad x^{1/4}>512,\qquad x^{3/10}>1024,$$

and therefore

$$1.94+\frac{1.03883}{64}+\frac{.94}{512} +\frac{1.03883}{1024} =\frac{200610011}{102400000}<1.96.$$

Hence

$$\boxed{\theta(x)>x-1.96\sqrt x \qquad(10^{11}\le x\le10^{19}).}$$

Together with the previously retained BFT bound below $10^{11}$, this extends $\theta(x)>x-2.072\sqrt x$ through $10^{19}$. This extension is particularly useful in the interval between the finite prime lookup and the eventual global analytic bound.

Broadbent et al., Corollary 2.1 also gives

$$\theta(x)\le(1+1.93378\cdot10^{-8})x\qquad(x\ge0),$$

and quotes $\theta(x)<x$ through $10^{19}$. The latter is used for pointwise bootstrap bounds; the former is a uniform upper bound for an unbounded tail.

Global errors and the finite-RH range

Broadbent et al., Table 14 bounds both sides of $\theta(x)-x$ for all $x>e^b$. The implementation includes its rows $b=20,25,30,35,40,45,50,60,70,80,90,100,200$. Every displayed error is increased by at least one final displayed unit: the paper explicitly states that the printed tables round their last digit. The thresholds $\lceil e^b\rceil$ are certified with directed intervals. For example, the rounded lower errors are

$$\begin{array}{c|cccccc} b&25&30&35&40&45&50\\\hline \delta_-(b)&7.4367\,10^{-6}&6.0752\,10^{-7}&4.9767\,10^{-8} &2.1483\,10^{-8}&1.1085\,10^{-8}&1.1345\,10^{-9}. \end{array}$$

Johnston–Yang, Lemma 2.2 provides

$$|\theta(x)-x|<\frac{\sqrt x\log^2x}{8\pi} \qquad(599<x\le2.169\cdot10^{25}).$$

This is a proved finite-range bound, obtained from the verified Riemann height; the helper does not assume the Riemann hypothesis outside that range. Since $\log^2x/\sqrt x$ decreases for $x>e^4$, its value at $x_0$ bounds the relative error up to the end of the verified range. Above that end, the global $b=50$ table row applies. Thus the maximum of these two errors is valid for every $x\ge x_0$. The square-root bound is patched at $10^{19}$ in the same way, using the global $b=40$ row.

The helper takes the minimum of all available valid envelopes. Square roots are bounded by integer square roots; logarithms use a directed 16-bin binary-mantissa table. Every subsequent calculation is rational.

The coefficient-gcd sum

Keep the intervals from the existing off-diagonal numerical argument:

$$U_t(m)=\frac{Hm-1}{t},\qquad V_t(m)=m\max\left\{ \frac{A+1/m_0}{k_A+1},\frac B{k_B+1},\frac{C+1/m_0}{k_C+1} \right\},$$

where $H=A+B+C$ and $k_X=\lfloor Xt/H\rfloor$. Only eligible $t$ with $k_A+k_B+k_C=t-2$ are used. The exact floor-stability check is unchanged. If $\epsilon_U,\epsilon_V$ are the new uniform errors above $U_t(m_0),V_t(m_0)$, respectively, then each interval contributes at least

$$m\left[(1-\epsilon_U)\frac{U_t(m_0)}{m_0} -(1+\epsilon_V)\frac{V_t(m_0)}{m_0}\right]$$

to the logarithm of either coefficient gcd. Negative contributions are replaced by zero and the others are rounded down before summing. This proves the returned lower rate for every $m\ge m_0$.

The sample certificates include directions $(17,11,26)$ with target $\ell=4.25708955$: $m_0=10^{12}$ and 10,000 terms give a strict rate margin exceeding $.0022149$, while the previous logarithmic Dusart helper at the same starting index misses the target by more than $.0033907$.

The same sample record certifies the three strengthened initial $2,7$, $3,7$, and $5,7$ sources with 100,000 terms and $m_0=10^{14}$; their previous retained uniform starts were $10^{40}$. These are analytic starting indices. Separate finite bootstraps may reduce them further.