John Change：Proof of the Analytical Equivalence of M(x, s) and the Riemann Zeta Function and Its Implications for the Riemann Hypothesis

 

Abstract

The Riemann Hypothesis (RH) is one of the most significant unsolved problems in analytic number theory. This paper constructs a new mathematical function M(x, s) and rigorously proves its analytical equivalence to the Riemann Zeta function ζ(s) across the entire complex plane, i.e., M(x, s) = ζ(s). This result provides new theoretical support for the Riemann Hypothesis. Using numerical methods such as Fourier transform, Laplace transform, and Mellin transform, we verify that M(x, s) exhibits the same zero structure as ζ(s). Through an analytical approach, we rigorously eliminate the error term O(x^{-s}), proving that M(x, s) analytically continues to ζ(s).

**Keywords:** Riemann Hypothesis, Analytic Number Theory, Zeta Function, Error Term, Analytical Continuation

**Mathematics Subject Classification (MSC):** 11M26, 11M06, 30D10

DOI：10.5281/zenodo.14955832

海之涛：基于新函数 M(x, s) 对黎曼 Zeta 函数的等价性证明及其对黎曼假设的影响

摘要

本文构造了一个新的数学函数 M(x, s)，并严格证明了其与黎曼 Zeta 函数 ζ(s) 在整个复平面上的解析等价性，即 M(x, s) = ζ(s)。这一证明为黎曼假设提供了新的理论支持。通过傅里叶变换、拉普拉斯变换、梅林变换等数值方法，我们验证了 M(x, s) 具有与 ζ(s) 相同的零点结构，并通过解析方法严格消除误差项 O(x^{-s})，从而证明 M(x, s) 在解析延拓后等于 ζ(s)。

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