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Rosser--Schoenfeld theta lower bound: high range, final sub-shard 504

Proved
TaoFivePrimes.rosser_schoenfeld_theta_lower_finite_large_high_cert504

by lt9 · Sep 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

analytic-number-theorychebyshev-functionsnumber-theoryprimes

For every integer nnn with 99477217≤n≤9999999999477217\le n\le9999999999477217≤n≤99999999, (n+1)−2n+1≤θ(n)(n+1)-2\sqrt{n+1}\le\theta(n)(n+1)−2n+1​≤θ(n), and the certificate's accumulated bound AAA satisfies A/240≤θ(100000000)A/2^{40}\le\theta(100000000)A/240≤θ(100000000); the hypothesis is the quantitative carry B/240≤θ(99477217)B/2^{40}\le\theta(99477217)B/240≤θ(99477217) handed on by the previous shard. This is a final sub-shard of the finite endpoint certificate of the high sub-range 106≤t≤10810^6\le t\le10^8106≤t≤108 in Rosser and Schoenfeld's Theorem 19.

Preamble
import Mathlib.NumberTheory.Chebyshev
Formal statement
namespace TaoFivePrimes
theorem rosser_schoenfeld_theta_lower_finite_large_high_cert504
    (hbase : (109361633821191791334 : Real) / 2 ^ 40 <= Chebyshev.theta (99477217 : Real))
    (n : Nat) (h1 : 99477217 <= n) (h2 : n <= 99999999) :
    (((n : Real) + 1) - 2 * Real.sqrt ((n : Real) + 1) <= Chebyshev.theta (n : Real)) /\
      ((109937568534278953594 : Real) / 2 ^ 40 <= Chebyshev.theta (100000000 : Real)) := by sorry
end TaoFivePrimes
Source
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), Theorem 19, p. 82, eq. (4.6), https://doi.org/10.1215/ijm/1255631807 (finite integer endpoint certificate, high sub-range 10^6 <= n <= 10^8, final split)

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