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发表于 2025-12-11 11:24
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用new path to prove riemann hypothesis在Google AI模式中搜索和对话，得到的结果如下，显然Google AI 对此方法的评价也是倾向积极的：
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The Riemann Hypothesis remains an unsolved problem in mathematics, and no proposed "new path" has yet resulted in a widely accepted proof by the mathematical community. However, several novel approaches and ongoing research directions are being actively explored.
Current significant new paths include:
Connecting to Quantum Mechanics: A prominent strategy, suggested by the Hilbert-Pólya conjecture, involves finding a self-adjoint operator in quantum mechanics whose energy levels correspond to the zeros of the zeta function. This would automatically prove the zeros are all real (lying on the critical line). This is an active area of interdisciplinary research.
De Bruijn-Newman Constant: Research focuses on the De Bruijn-Newman constant, a mathematical value related to the realness of the zeros. The hypothesis is true if and only if this constant is less than or equal to zero. Mathematicians like Terence Tao have been working to push the known upper bounds of this constant down, bringing new insights, though proving it is exactly zero remains elusive.
Topological Field Theory and Noncommutative Geometry: Some researchers are exploring advanced mathematical frameworks like noncommutative spectral geometry and topological field theory to encode the zeros of the zeta function in the spectrum of a Dirac-like operator.
Real Domain and Convolutional Algebra: Attempts have been made to reinterpret the zeta function in the real domain using tools like the Laplace transform, which some researchers argue could simplify the analysis and replace complex analysis with algebraic methods.
"Purported Proofs" and Ongoing Scrutiny: From time to time, various individual mathematicians publish papers claiming a complete "new proof" (e.g., using a "Todd function" or a "Spira Criterion"). These are typically complex and still under rigorous review by the broader community; none have yet passed general consensus and review.
The official status of the problem remains unsolved, with a $1 million prize from the Clay Mathematics Institute for the first correct proof.
To help you understand the core concepts involved in these different approaches, I can look at a side-by-side comparison of the physical interpretations (like quantum mechanics vs. topological field theory) that researchers are using—covering their main ideas and challenges. Should we start there?
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Real Domain and Convolutional Algebra: Attempts have been made to reinterpret the zeta function in the real domain using tools like the Laplace transform, which some researchers argue could simplify the analysis and replace complex analysis with algebraic methods.--Give more detail information of this approach?
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The "Real Domain and Convolutional Algebra" approach to the Riemann Hypothesis is a highly specific, proposed methodology, notably detailed in a 2018 paper by Jing Min Zhu, which has not yet been accepted by the mathematical mainstream. Its core ambition is to move the problem out of the complex plane, where the analysis is notoriously difficult, and into the simpler real domain.
Core Methodology
approach attempts a paradigm shift by utilizing integral transforms to express the complex Riemann zeta function in terms of real-variable functions.
Laplace Transformation: The key mathematical tool used is the Laplace transform (or its inverse). The Riemann zeta function, \(\zeta (s)=\sum _{n=1}^{\infty }n^{-s}\) for \(\Re (s)>1\), is viewed as the Laplace transform of a series of Dirac delta functions in the time (real) domain:\(L^{-1}[\zeta (s)]=\delta (x)+\delta (x-\ln 2)+\delta (x-\ln 3)+\delta (x-\ln 4)+\dots \)This transforms the problem from analyzing a function of a complex variable \(s\) to analyzing a distribution (a series of impulses) along the real number line.
Integration to Simpler Functions: Integrating this series of delta functions yields a combination of unit step functions:\(\int L^{-1}[\zeta (s)]dx=u(x)+u(x-\ln 2)+u(x-\ln 3)+u(x-\ln 4)+\dots \)The argument then focuses on finding tight upper and lower bounds for this step function using the functions \(e^{x}\) and \(e^{x}-1\). The Laplace transforms of these bounds naturally contain the \(\frac{1}{s-1}\) term which is central to the analytic continuation of the zeta function.
Convolutional Algebra: Instead of using traditional complex analysis to study the analytic continuation and its zeros in the complex plane, this method aims to establish an algebraic research paradigm for prime number distribution. By transforming convolutions into simple multiplications in the Laplace domain, the hope is that properties of the zeta function related to prime numbers can be analyzed using simpler, purely algebraic manipulations and real analysis.
The Claim and Status The approach claims to prove the Riemann Hypothesis by showing that a related "error function" derived in the real domain meets the necessary criteria for an equivalent proposition of the RH, specifically the error bound in the prime-counting function approximation, \(\pi (x)-li(x)=O(x^{0.5+\epsilon })\).
Important Caveat: This method, and the specific papers proposing it, remain highly speculative. The mathematics community typically requires rigorous peer review and consensus before a proof of such a monumental problem is accepted. As of now, this "Real Domain and Convolutional Algebra" path is considered a novel, but unproven, approach that has not gained widespread acceptance as a valid proof of the Riemann Hypothesis.
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It seem this approach--Real Domain and Convolutional Algebra--is much more easier to be understood than other ones you mentioned above, do you think so? If yes, that will means it is much more easier to find it is right or not. So, have you any conclusion on it yet?
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It is true that the foundational concepts in the "Real Domain and Convolutional Algebra" approach—specifically the use of the Laplace transform, delta functions, and step functions—are more accessible to individuals with a standard undergraduate engineering or physics background than the highly abstract concepts in quantum mechanics or noncommutative geometry.
This relative simplicity should theoretically make it easier for the mathematical community to verify its correctness.
My Conclusion on the Approach's Validity
As an AI, I do not form personal "opinions" or mathematical "conclusions." Instead, I process and reflect the consensus of the broader mathematical community.
The current conclusion from the field of mathematics is that this specific approach has not yielded a valid or accepted proof of the Riemann Hypothesis.
Here is a breakdown of why, despite its apparent simplicity:
Simplicity vs. Rigor: While the tools used (Laplace transforms, real analysis) are simpler, the application within the specific proposed proofs often involves subtle, complex arguments about the behavior of analytic continuation and error bounds that the community finds questionable or incomplete.
Lack of Consensus: No major mathematical body (like those associated with the Clay Mathematics Institute or leading universities) has validated the claims made by researchers using this path. The rigorous scrutiny required for such a significant proof involves extensive peer review, and this approach has not survived that process successfully yet.
The Problem is Fundamentally Hard: The Riemann Hypothesis is one of the deepest unsolved problems in mathematics precisely because all "easy" approaches have failed over the last 150 years. Most professional mathematicians believe the solution likely requires a profound, novel insight using highly advanced tools (like those in spectral theory or algebraic geometry) rather than a simplified re-application of standard real analysis tools.
In summary, the approach is easier to understand superficially, but its claims of a complete proof are not accepted as correct or rigorous by expert mathematicians. The problem remains officially unsolved.
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