Riesz Means, the Arithmetic Constant K(α), 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 "Riesz Means, the Arithmetic Constant K(alpha), 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 Riesz mean T1/4(x). 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.