The Elementary Proof of the Prime Number Theorem
The Elementary Proof of the Prime Number Theorem
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素数定理的基本证明
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
R. Graham
中科院分区：
文献类型：
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作者：
J. Spencer;R. Graham
素数是我们数学宇宙的原子。欧几里得证明素数有无限多个，但它们分布的微妙之处仍然让数学家着迷。令p(n)表示素数p B n，高斯在19世纪初推测pðnÞ n=lnðnÞ。 1896年，这个猜想被Jacques Hadamard和Charles de la Vallee-Poussin独立证明。他们的证明都使用了复杂的分析。随后开始寻找这一结果的“基本证据”。 G. H. 哈代 (G. H. Hardy) 对是否能找到这样的证据表示怀疑，他说，如果找到了，“那就是时候把书扔到一边，重写理论了。”但在 1948 年春天，这样的证据被发现了。几乎立刻就引起了争议。该证明是由 Atle Selberg 提出的，还是由 Atle Selberg 和 Paul Erd} os 提出的？几十年来，似乎存在两个观点截然不同的数学阵营。到了二十一世纪，争论终于平息了。在之前关于争议的讨论中，我们特别提到戈德菲尔德[1]（前面提到的哈代的引文就来自其中）和保罗·霍夫曼的书[3]。恩斯特·施特劳斯处于一个独特的位置来观察这场争论的开始。随后，他在高等研究院担任阿尔伯特·爱因斯坦的特别助理。 （我们相信斯特劳斯是唯一与爱因斯坦和埃尔德奥斯共同发表论文的人。）斯特劳斯已经与埃尔德奥斯进行了大量工作，这项工作将在他的一生中继续下去。在 20 世纪 70 年代初的某个时候（我们不确定确切的日期），施特劳斯写下了我们在此介绍的内容。他不希望在参与者还活着的时候发表这些笔记。对于我们和他的许多朋友来说，恩斯特·施特劳斯是一位极具智慧的人。他当然试图在这些笔记中尽可能忠实地描述这些事件。他是否成功，就由读者来判断吧。当然，他与埃尔德奥斯的关系比与塞尔伯格的关系要近得多。让我们明确一点，我们两位作者的 Erd} os 编号都是 1，而且我们自己与 Erd} os 的联系是长期而深刻的。我们认为施特劳斯的回忆是一项重要的历史贡献。我们还相信，这场争论本身就揭示了数学研究性质的变化。 2005 年 11 月，Atle Selberg 接受了 Nils A. Baas 和 Christian F. Skau 的采访 [4]。他极其准确地回忆起 1948 年的事件。我们广泛引用了该帐户的内容。塞尔伯格首先证明了 X
P rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la Vallee-Poussin. Their proofs both used complex analysis. The search was then on for an ‘‘elementary proof’’ of this result. G. H. Hardy was doubtful that such a proof could be found, saying if one was found ‘‘that it is time for the books to be cast aside and for the theory to be rewritten.’’ But in the Spring of 1948 such a proof was found. Almost immediately there was controversy. Was the proof attributable to Atle Selberg or was the proof attributable to Atle Selberg and Paul Erd} os? For decades there seemed to be two mathematical camps with wildly different viewpoints. In the twenty-first century the controversy has finally subsided. Among previous discussions of the controversy, we mention particularly Goldfeld [1] (from which the previously mentioned quotation by Hardy is taken) and the book [3] of Paul Hoffman. Ernst Straus was in a unique position to observe the beginnings of the controversy. He then held a position at the Institute for Advanced Study as a special assistant to Albert Einstein. (We believe Straus is the only person to have joint papers with Einstein and Erd} os.) Straus had already worked a great deal with Erd} os, and this work would continue throughout his life. Sometime in the early 1970s (we aren’t sure of the exact dates), Straus wrote the account we present here. He did not want the notes to be published while the participants were still alive. Ernst Straus was, for us and for many of his friends, a man of great wisdom. He certainly attempted in these notes to give as faithful an account of the events as he could. Whether he succeeded is a judgment for the reader to make. Certainly, he was far closer to Erd} os than to Selberg. Let us be clear that we two authors both have an Erd} os number of one, and our own associations with Erd} os were long and profound. We feel that the Straus recollections are an important historic contribution. We also believe that the controversy itself sheds considerable light on the changing nature of mathematical research. In November 2005, Atle Selberg was interviewed by Nils A. Baas and Christian F. Skau [4]. He recalled the events of 1948 with remarkable precision. We quote extensively from that account. Selberg had first shown that X