Rosser-Schoenfeld theta lower bound beyond 10^10

The Rosser–Schoenfeld lower bound for the Chebyshev $\theta$ function, in the range beyond numerical verification:

Why this range is separate. The mission already has the companion statements TaoFivePrimes.rosser_schoenfeld_theta_lower_analytic_mid for $1420 \le t \le 10^{10}$ and TaoFivePrimes.rosser_schoenfeld_theta_lower_finite for $255 \le t \le 1420$ (in the form $t - 2\sqrt{t} < \theta(t)$). Together with this statement they cover $t \ge 1340$ with no gap, which is what TaoFivePrimes.rosser_schoenfeld_theta_lower_analytic asserts; without it that node cannot be closed, since the $\le 10^{10}$ ceiling on the middle range is exactly where tabulated data stops.

What the proof needs. This is the range where the estimate stops being a computation and becomes analysis: the bound follows from the explicit formula together with a zero-free region for $\zeta$, in the form Rosser and Schoenfeld give (Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), Theorem 10 and its refinements). The error term $t/(2\log t)$ is far larger than the truth — $\theta(t) = t + O(t\exp(-c\sqrt{\log t}))$ — so the statement has enormous room, and a crude zero-free region suffices. What it cannot be is a finite check.

The threshold $10^{10}$ is a seam, not a feature. Any constant between $1420$ and the point where the analytic argument starts working would do; $10^{10}$ is chosen to match the ceiling of the middle-range node so that the three ranges abut exactly. In particular there is no claim that $10^{10}$ is where the analytic method first applies.

Note also that the $\sqrt{t}$ form of the finite range and the $1/(2\log t)$ form here are comparable only for $\sqrt{t} \ge 4\log t$, which holds from $t \ge 1296$ onward — that is why the assembled statement starts at $1340$ rather than at $255$.