Abstract
This paper presents a computational framework for the rigorous numerical verification of the Riemann Hypothesis (RH) through advanced algorithmic techniques.
Computational Verification and the Riemann Hypothesis
This paper advances the numerical verification of the Riemann Hypothesis through the development of high-precision algorithms for isolating zeros of the Riemann zeta function ζ(s) on the critical line Re(s) = 1/2. Unlike approaches that seek operator-theoretic or spectral realizations, this work operates within the computational paradigm, treating the RH as a problem of rigorous algorithmic complexity and error-controlled numerical analysis. The mathematical objects at the core are the Riemann-Siegel Z-function, Gram points, and Lehmer pairs, analyzed through the lens of Turing's method for explicit zero counting.
Algorithmic Innovations
The central contribution is a hybrid zero-isolation algorithm that achieves complexity O(T^{3/4+ε}) for verifying all zeros up to height T, improving upon the traditional O(T^{1+ε}) bounds of direct evaluation methods. This is accomplished by integrating the fast multipole method (FMM) for accelerating oscillatory sums with rigorous ball arithmetic to bound floating-point errors. The paper establishes explicit constants for the remainder terms in the Riemann-Siegel formula, allowing for machine-verified proofs of zero location without relying on heuristic error estimates.
Rigorous Error Analysis
A key theoretical result provides computable error bounds for floating-point evaluations of ζ(1/2 + it) at large t. Using techniques from interval arithmetic and arbitrary-precision ball arithmetic, the authors prove that their algorithm maintains certified accuracy even when t exceeds 10^20. This addresses a fundamental challenge in computational number theory: the gap between numerical evidence (which suggests RH is true) and rigorous verification (which requires proof that rounding errors cannot create spurious zeros or miss actual ones).
Numerical Results and Implications
The computational framework verifies that the first N zeros (where N corresponds to heights T ≈ 10^21) satisfy the hypothesis, extending the previous record by Platt (T ≈ 3 × 10^19). The paper includes Wolfram Language implementations that demonstrate the practical feasibility of these algorithms. By pushing numerical verification into these extreme heights, the work tests the asymptotic behavior of the zeta function in regimes where the GUE (Gaussian Unitary Ensemble) statistics of zero spacing should dominate, providing empirical validation of random matrix conjectures while simultaneously strengthening the evidential basis for the Riemann Hypothesis itself.
Conjectures on Algorithmic Limits
The paper concludes with a conjecture regarding the algorithmic verification threshold, proposing that the maximum height T_max achievable with computational resources growing exponentially according to Moore's Law satisfies T_max ~ exp(c √N_ops), where N_ops is the number of operations. This suggests a pathway by which computational verification could eventually reach heights relevant to the De Bruijn–Newman constant or other critical-line phenomena, potentially revealing subtle violations of RH if they exist at ultra-large scales.