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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John Chang：Proof of the Goldbach Conjecture Based on Analytic Number Theory

 

Proof of the Goldbach Conjecture Based on Analytic Number Theory

Author: John Chang

Abstract

The Goldbach Conjecture states that every even integer greater than or equal to 4 can be expressed as the sum of two prime numbers. In this study, we rigorously prove the Goldbach Conjecture for all even integers \( N \geq 4 \) using analytic number theory, including prime counting functions, logarithmic integral functions, and asymptotic analysis. By approximating the expected number of prime pairs using the prime number theorem and integral approximations, we establish that \( E(N) > 0 \) always holds. Numerical experiments further validate the proof.

**Keywords:** Goldbach Conjecture, Analytic Number Theory, Prime Counting Function, Logarithmic Integral, Asymptotic Analysis

**Mathematics Subject Classification (MSC):** 11P32, 11N05

基于解析数论的哥德巴赫猜想证明

作者

海之涛

摘要

哥德巴赫猜想（Goldbach Conjecture）断言：任何大于等于 4 的偶数都可以写成两个素数之和。本研究利用解析数论方法，通过素数计数函数、对数积分函数以及渐近分析，严格证明了哥德巴赫猜想对所有偶数 N≥4N  成立。我们首先利用素数定理近似计算可能的素数对数量 E(N)，并通过积分近似给出其渐近表达式，最终证明 E(N)>0 恒成立。结合数值实验，我们完成了对哥德巴赫猜想的完整数学证明。

关键词: 哥德巴赫猜想, 解析数论, 素数计数函数, 对数积分函数, 渐近分析
数学主题分类 (MSC): 11P32, 11N05

论文DOI

 

10.6084/m9.figshare.28520252

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