The Arithmetic Constant K(α), a Stabilised Divisor Sieve and Numerical Evidence for the Dirichlet Divisor Conjecture — Code and Data

This repository contains all source code, numerical data, and the preprint for the paper "The Arithmetic Constant K(α), a Stabilised Divisor Sieve and Numerical Evidence for the Dirichlet Divisor Conjecture" by Ilias A. Pappas (2026). The paper studies F(x) = Delta(x)/x1/4, the normalised error term in the Dirichlet divisor problem, via the weighted divisor sum T1/4(x)=Σd(n)/n^1/4. Theoretical contributions include a new closed-form arithmetic constant K(1/4) = zeta(1/4)^2 - 4/9 + (2*gamma-1)/3 ≈ 0.2685, and a complete equivalence between the divisor conjecture theta=1/4 and the boundedness of E(x) = T1/4(x) - M_D(x,1/4) - K(1/4). Computational contributions include an exhaustive Kahan-stabilised parallel sieve computing max|F(x)| for all integers in [10^8, 10^13] -- the first such computation beyond 10^10. Statistical analysis of five decades shows the Huxley-normalised ratio max|F|/x27/416 decreasing monotonically from 1.773 to 1.262, placing the Huxley bound theta=131/416 outside the 99% confidence interval, consistent with the divisor conjecture theta=1/4. Contents: C++ source code (exact_E8.cpp, sample_E.cpp), all numerical output files (s*.txt, p*.txt, sample*.txt), and the paper PDF. *Version 2 adds distributional analysis over six decades [10^8, 10^14], including moment statistics, KS test (sample_E_parallel.cpp, ks_test.cpp), and sampling data for [10^13, 10^14].*