v1.11 — A Numerical Study of the Division Table

What is new

papers/a_numerical_study_of_the_division_table.md — the most elementary paper in the repository, and the natural entry point. It starts from the quotients n/d for n ≤ 100 written as mixed numbers, and reports at every step which classical object the sheet has become.

Three features before any theory: the whole cells are the summatory divisor function D(x), the column sums are the harmonic numbers, and the determinant of the 0–1 shadow — after one column is falsified — is the Mertens function, so det Rₙ = O(n^{1/2+ε}) is the Riemann hypothesis wearing a matrix. The paper says plainly that this is the hypothesis, not a route to it.

Then a walk across the table traces an arch; the arch is Fermat's method (1643); its failure is repaired by Kraitchik and the quadratic sieve, worked by hand on N = 2041; and rotating the picture forty-five degrees lands on the Dirichlet divisor and Gauss circle problems, Voronoï's series, and the x^{1/4} they share — which the paper then dissolves by softening the edge, leaving the single explicit term −1/(144X).

Two closing sections truncate the sheet instead of walking it. Keeping only the rows up to a cutoff z turns the table into the sieve of Eratosthenes and makes primality a time-dependent verdict, read on the clock u = log x / log z: Buchstab's ω as the exact dual of Dickman's ρ from §5, the frontier u = 2 where every survivor is prime, and p² as the same statement in other coordinates. The last section reduces a family of measurements to one identity, A_p = p·S_p⁻, and ends on a hard limit: the mean quantities of the sieve settle inside a window of 10⁶, while the extreme ones need a complete cycle whose length is a primorial — the largest gap among the odd numbers coprime to every odd prime up to 53 is 106, and the best a scan to 10⁹ shows is 64.

No result in the paper is new, and it says so in the abstract. Its ledger section lists twenty-one items and decomposes all twenty-one into named classical results.