A Proof of the Riemann Hypothesis

The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, conjectures that all non-trivial zeros of the Riemann zeta function lie on the critical line ℜ ( s ) = 1 2 . This hypothesis is deeply connected to the structure of prime numbers and has profound implications in mathematics and physics. In this work, we present a proof grounded in symmetry and logical reasoning, deriving the zeta function from interconnected mathematical principles such as Euler’s product formula, Fourier analysis, and modular forms. We establish that the symmetry of the zeta function about the critical line is an inherent property arising from its analytic continuation and functional equation. Additionally, a proof by contradiction demonstrates that any deviation from this symmetry results in inconsistencies across multiple mathematical domains. This logical framework affirms the inevitability of the hypothesis, offering a unified perspective based on established mathematical truths. Supplementary Material File (reimann_hypothesis.pdf) - Download - 266.91 KB Information & Authors Information Version history Copyright This work is licensed under a Non Exclusive No Reuse License.

Authors Metrics & Citations Metrics Article Usage 1120views 168downloads Citations Download citation Abdul Musawir. A Proof of the Riemann Hypothesis. Authorea. 13 February 2025. DOI: https://doi.org/10.22541/au.173945714.49665647/v1 DOI: https://doi.org/10.22541/au.173945714.49665647/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu.