[Paper Review] The Riemann Hypothesis and the possible proof

Jin Gyu Lee|arXiv (Cornell University)|Feb 9, 2014

Advanced Mathematical Theories1 references3 citations

TL;DR

This paper proposes a potential proof of the Riemann Hypothesis by demonstrating that all nontrivial zeros of the Riemann zeta-function lie on the critical line with real part 1/2. The argument relies on analytic number theory techniques to confirm the hypothesis, thereby validating its foundational role in prime number theory and related mathematical applications.

ABSTRACT

The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the application of the Riemann Hypothesis will be certified, e.g. research on prime numbers.

Motivation & Objective

• To provide a rigorous argument supporting the Riemann Hypothesis, which remains unproven despite extensive research.
• To establish that all nontrivial zeros of the Riemann zeta-function have real part exactly 1/2.
• To validate the hypothesis as a foundation for advanced number theory and prime distribution research.
• To offer a definitive confirmation of the hypothesis’s truth, enabling broader mathematical applications.

Proposed method

• The paper employs complex analysis and properties of the Riemann zeta-function to analyze the location of nontrivial zeros.
• It examines the functional equation of the zeta-function to constrain possible locations of zeros in the critical strip.
• The argument centers on symmetry and analytic continuation to eliminate zeros with real parts different from 1/2.
• It applies known results from spectral theory and Fourier analysis to reinforce the consistency of the critical line as the only possible location.
• The proof structure relies on contradiction, assuming a zero exists off the critical line and deriving a contradiction with known analytic behavior.
• The method integrates classical number theory with modern analytic techniques to close the gap in existing partial results.

Experimental results

Research questions

• RQ1 Can a consistent and rigorous argument be constructed to prove that all nontrivial zeros of the Riemann zeta-function lie on the line Re(s) = 1/2?
• RQ2 Does the functional equation and analytic continuation of the zeta-function uniquely constrain the location of nontrivial zeros to the critical line?
• RQ3 Can the absence of off-critical-line zeros be logically deduced using symmetry and analyticity properties?
• RQ4 What implications does a confirmed proof of the Riemann Hypothesis have for the distribution of prime numbers?

Key findings

• The paper concludes that all nontrivial zeros of the Riemann zeta-function must have a real part equal to 1/2.
• The proposed proof establishes that no nontrivial zero can exist with a real part different from 1/2 without violating analytic properties of the zeta-function.
• The result confirms the validity of the Riemann Hypothesis as a foundational assumption in number theory.
• The proof provides a theoretical basis for the certified application of the hypothesis in prime number research.