FALSEHOOD OF THE RIEMAN HYPHOTHESIS

Note on this version: This paper is a revised and updated version of my paper published on ResearchGate in December 2025 (DOI: [10.13140/RG.2.2.18216.02565]). This version includes important corrections regarding the Riemann Hypothesis. My previous paper published on ResearchGate contained a significant error, and for a long time I mistakenly believed that limn→∞logPn/logn=1 was incorrect.I have since proven that this limit is actually correct (similarly for Pn ~ n log n). I deeply apologize for my oversight. I sincerely apologize for any inconvenience caused to my readers. In this revised version, I accept that limn→∞logPn/logn=1 (also for Pn ~ n log n) holds, and after carefully examining the distribution of prime numbers, I have concluded that the Riemann Hypothesis is false. This is a very meticulously analyzed paper, and I am confident that its content is compelling. Please take a look. 【Abstract】This paper examines the asymptotic formula for prime numbers Pn = n{logn+loglogn+O(1)} and the error term O(1). Considering Littlewood’s theorem, the error term O(1) must be composed of two formulas: Rosser, Schoenfeld’s formula and P. Dusart, Ch.Axler’s formula. Both formulas cannot be used simultaneously; prime numbers must be contained within one of these formulas. This is what has made solving the Riemann Hypothesis so difficult. And, we will consider the theorem by H.von Koch and show the falsehood of the Riemann hypothesis.本版本说明：本文为笔者2025年12月发布于ResearchGate的论文的修订更新版（DOI：[10.13140/RG.2.2.18216.02565]）。本版本包含关于黎曼猜想（Riemann Hypothesis）的重要修正内容。 笔者此前发布于ResearchGate的论文存在一处重大错误：长期以来笔者曾错误认为limₙ→∞(logPₙ/logn)=1这一极限式不成立。后续笔者已证明该极限实际成立（同理，Pₙ~n logn也成立）。笔者为自身的疏忽深表歉意，并为给读者带来的不便致以诚挚的歉意。 在本次修订版中，笔者确认limₙ→∞(logPₙ/logn)=1成立（Pₙ~n logn亦成立）；在仔细研究素数分布后，笔者得出黎曼猜想（Riemann Hypothesis）不成立的结论。本文经过极为细致的分析论证，笔者坚信其内容具有说服力，恳请各位读者参阅。 【摘要】本文针对素数的渐近公式Pₙ = n{logn + loglogn + O(1)}及其误差项O(1)展开研究。结合李特尔伍德定理（Littlewood’s theorem），误差项O(1)实际由两类公式构成：罗斯-肖恩费尔德公式（Rosser, Schoenfeld’s formula）与迪萨尔-阿克塞尔公式（P. Dusart, Ch.Axler’s formula）。二者不可同时使用，素数分布必然仅符合其中一类公式，这也是黎曼猜想（Riemann Hypothesis）的证明长期陷入困境的原因所在。此外，本文将借助H·冯·科赫定理（H.von Koch theorem）论证黎曼猜想不成立。