Atle Selberg (1917–2007) was one of the titans of contemporary number theory, known, for example, for the so-called elementary proof of the prime number theorem and of course for the trace formula that bears his name.
The prime number theorem’s elementary proof is notorious for being ridden with controversy, with many believing that the Fields medal awarded to Selberg in 1950 should have been shared with Paul Erdős. But it wasn’t, of course. The details of the affair, in Selberg’s own words, can be found in http://www.ams.org/journals/bull/2008-45-04/S0273-0979-08-01223-8/home.html, which transcribes an in-depth interview with Selberg done in 2005 by Nils Baas and Christian Skau. The particular difficulties and controversies surrounding priority regarding the elementary proof of PNT, or, more precisely, the degree to which Erdős was involved in the proof as it finally appeared in print, are presented in great detail in this article, and it makes for both interesting reading (even to non-number theorists) and a record of a very unfortunate chain of events and misunderstandings. Perhaps the easiest way to describe it all is that Erdős was a natural collaborator whereas Selberg was an equally natural loner. The famous Strother Martin line comes to mind: “What we have here is a failure to communicate …” (Cool Hand Luke, Warner Bros-7 Arts, 1967).
A personal sidelight on the Erdős-Selberg business: I first learned number theory from the late Ernst Straus in the middle 1970s, and in our third quarter he presented the elementary proof of PNT as something of an expert eye-witness account (and quite a tour-de-force): as the above article indicated, Straus, one of Erdős’ close friends and a frequent collaborator and co-author, was at the Institute for Advanced Study when Selberg and Erdős were, so to speak, going at it. That school year (I think it was 1975) Erdős came to visit — all very impressive stuff, and formative for me, really, as my undergraduate days wore on. However, in graduate school I did my thesis with Audrey Terras, much of whose work was focused on applications of Selberg’s trace formula, which engendered, for me, a definitive departure from elementary number theory to, broadly speaking, automorphic forms. But now Selberg’s presence was undeniable, and it was then, in the early 1980s, as a rube graduate student, that I truly began to take note of him. He was certainly one of Audrey’s heroes and that naturally made a huge impression.
The book under review, a reprint of the 1989 edition, presents a trajectory of Selberg’s work including his papers on PNT (as well as the Dirichlet theorem on primes in arithmetical progressions, also done “elementarily”, i.e., without complex analysis), as well as the trace formula. The years covered are 1936–1988; volume II deals with the papers of a later date. The book under review also contains a great deal of material on the Riemann Hypothesis, and of course material on sieve methods, another area in which Selberg enjoyed a tremendous reputation.
There are a number of articles in German, and a couple in Norwegian (including an account of correspondence with Brun, Jacobsthal, and (yes! Carl Ludwig) Siegel about the Riemann zeta function), but most are in English. I can only repeat the great advice given by another Norwegian, Niels Henrik Abel: “It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils.” And Selberg was a very great master.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.