Numerical Behavior of the Riemann Zeta Function Using Real-to-Complex Conversion Without Gram Points or Bracketing

TL;DR

This paper introduces a novel real-to-complex parametrization method for analyzing the Riemann zeta function, enabling high-precision zero computations up to height 1e20 without classical Gram point bracketing.

Contribution

It develops a new numerical framework that replaces classical methods with a real-to-complex approach, improving zero detection and analysis of the zeta function.

Findings

Zero computations up to height 1e20 performed

Observed zero spacing matches Riemann-von Mangoldt predictions

Method demonstrates high accuracy and reproducibility

Abstract

The Riemann zeta function and the distribution of its nontrivial zeros on the critical line remain central topics in analytic number theory and large-scale computation. This work develops a numerical framework that replaces classical Gram-point bracketing with a real-to-complex parametrization of the critical line. Combined with high-precision evaluation of the Hardy Z function using the Riemann-Siegel formula with Gabcke type remainder control, this parametrization induces a Valley Scanner algorithm that identifies local minima of the absolute value of Z and refines them using safeguarded Newton iterations. The method exploits the mountain-valley geometry of the Z function rather than sign changes, and is implemented in a cloud based C language and MPFR pipeline with parallel execution on multi-core CPU instances. The paper reports Zero computations from the classical low lying range up to heights near 1e20, compares the observed spacing of zeros with Riemann - von Mangoldt predictions across several test windows, and summarizes consistency checks based on independent datasets and a local ball test. The work also introduces a fully real staircase formulation of the explicit-formula correction term for the Chebyshev function, expressed directly in terms of the zero ordinates and derived quantities. All numerical datasets, Docker images, and reproducibility materials are publicly archived via Zenodo and GitHub.