The Elementary Proof of the Prime Number Theorem

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 Vallée-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