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) /x^1/4, 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) - MD (x, 1/4) - K (1/4). Computational contributions include an exhaustive Kahan-stabilised parallel sieve computing max|F (x) | for all integers in 10⁸, 10¹3 -- the first such computation beyond 10¹0. Statistical analysis of five decades shows the Huxley-normalised ratio max|F|/x^27/416 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 (exactE8. cpp, sampleE. cpp), all numerical output files (s*. txt, p*. txt, sample*. txt), and the paper PDF.