Prove Riemann Hypothesis (리만가설을 증명해봐)

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Proving the Riemann Hypothesis is one of the most significant unsolved problems in mathematics, particularly in the field of number theory. As of my last update, the Riemann Hypothesis remains an open problem, and no proven solution has been accepted by the mathematical community.

The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is a conjecture about the zeros of the Riemann zeta function, which is defined as follows for complex numbers (s) with real part greater than 1:

[

\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}

]

The function can also be analytically continued to other values of (s) (except (s = 1), where it has a simple pole) through various methods, including the Euler product formula:

[

\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^s}\right)^{-1}

]

The Hypothesis:

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function have their real parts equal to ( \frac{1}{2} ). That is, any zero ( \rho ) of ( \zeta(s) ) that is not a negative even integer (these trivial zeros are easily verified by the functional equation of the zeta function) satisfies:

[

\text{Re}(\rho) = \frac{1}{2}

]

Why It Matters:

The importance of the Riemann Hypothesis lies in its implications for the distribution of prime numbers. The accuracy of the prime number theorem, which estimates the number of primes less than a given number, improves under the assumption that the Riemann Hypothesis is true. Specifically, the hypothesis is intimately connected to the error term in the prime number theorem, and proving it would lead to more precise information about the distribution of prime numbers.

Current Approaches and Challenges:

Analytical Number Theory: Techniques involving complex analysis, Fourier analysis, and the study of ( L )-functions (generalizations of the zeta function) are commonly used. Many results conditional on the Riemann Hypothesis have been proven, showcasing potential consequences were it to be resolved.

Random Matrix Theory: Insights from physics have suggested analogies between the zeros of the zeta function and the eigenvalues of random matrices. This connection has been fruitful in providing statistical properties of the zeros.

Computational Checks: High-precision computations have verified that the first 10 trillion zeros of the zeta function (in the critical strip where the real part is between 0 and 1) lie on the critical line ( \text{Re}(s) = \frac{1}{2} ), supporting the hypothesis but not proving it in general.

Other Zeta Functions: Studies on function fields and the zeta functions of curves over finite fields have successfully proven analogues of the Riemann Hypothesis, lending hope that similar techniques might apply to the classical setting.

Proving the Hypothesis:

A proof of the Riemann Hypothesis would likely require a novel insight into the nature of the zeta function, potentially involving new mathematics that could bridge existing gaps in understanding. Any proposed proof would need to be rigorously peer-reviewed and would involve advanced techniques from several areas of mathematics.

In conclusion, while the Riemann Hypothesis is a central unsolved puzzle in mathematics, its resolution—whether proving it true or false—would be a monumental achievement with deep repercussions across number theory, complex analysis, and related fields. If you are interested in exploring this topic further, I recommend studying complex analysis, analytic number theory, and the theory of ( L )-functions.

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여태 걍 "리만가설 증명해봐" 하면 다들 걍 리만가설의 역사나 정의만 나불거리다가 인공지능 챗봇으로서 난 못한다 하고 엄청 짤막하게 땡 쳤는데

이놈은... 다른 모델한테

엄청 장문의 프롬프트... "할 수 없다가 아니라 여러 가지로 분석하고 고민해보고 뭐뭐뭐뭐.... 해서 어쩌구저쩌구" 라는 부가적인 프롬프트 엔지니어링을 해야지만 그제야 이것저것 더 설명해주는 걸 그런 거 필요 없이 한방에 해내주는 느낌 ㄷ