基于交叉熵优化与推理的黎曼猜想分析
Analysis on Riemann Hypothesis with Cross Entropy Optimization and Reasoning
AI总结：
本文提出一种结合交叉熵优化、大数定律和数学归纳法的框架分析黎曼猜想，并引入基于大语言模型多路径思维链累积概率的增强推理方法，为最终证明黎曼猜想提供新思路。
摘要：
本文提出了一个用于分析黎曼猜想[27]的新框架，该框架由三个关键部分组成：a) 基于交叉熵优化与推理的概率建模；b) 大数定律的应用；c) 数学归纳法的应用。分析主要通过交叉熵优化与推理的概率建模以及稀有事件模拟技术进行。大数定律[2, 3, 6]的应用和数学归纳法的应用使得对黎曼猜想的分析自洽且完整，从而确保整个复平面如黎曼猜想所推测的那样被覆盖。我们还讨论了利用大语言模型（LLMs）进行推理的增强 top-p 采样方法，其中下一个 token 的预测不仅基于当前轮次中每个可能 token 的估计概率，还基于多个 top-k 思维链（CoTs）路径之间的累积路径概率。交叉熵优化与推理的概率建模可能非常适合黎曼猜想的分析，因为黎曼 Zeta 函数本质上处理的是复数级数中无限分量之和。我们希望本文的分析能够为黎曼猜想的一些洞见提供启示。本文提出的框架和技术，结合大语言模型（LLMs）中思维链（CoT）或思维图（DoT）推理与强化学习（RL）[1, 7, 18, 21, 24, 34, 39-41]的最新进展，可能为最终证明黎曼猜想[27]铺平道路。
英文摘要：
In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].
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