Reflections on Paul Erd˝os on His Birth Centenary Krishnaswami Alladi and Steven Krantz, Coordinating Editors This is Part I of a two-part feature on Paul Erd˝osfollowing his centennial. There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics. Here in Part I we have contributions from Krishnaswami Alladi and Steven Krantz, László Lovász and Vera T. Sós, Ronald Graham and Joel Spencer, Jean-Pierre Kahane, and Mel Nathanson. Part II will contain articles by Noga Alon, Dan Goldston, András Sárközy, József Szabados, Gérald Tenenbaum, and Stephan Garcia and Amy Shoemaker. Krishnaswami Alladi and Steven Krantz One of the Most Influential Mathematicians of Our Time The 100th birth anniversary of the great Hun- garian mathematician Paul Erd˝oswas celebrated in Budapest in July 2013 with an international conference that attracted about 750 participants. Erd˝oswas one of the most influential mathemati- cians of the twentieth century for a variety of reasons. He made fundamental and pioneering contributions in several fields of mathematics, such as number theory, combinatorics, graph theory, analysis, geometry, and set theory. He was perhaps the most prolific mathematician in history after Euler, with more than 1,500 papers, but what was most interesting about this was that more than half of these were joint papers and many of his collaborators were very young. Thus through the fundamental ideas in his papers and through these Courtesy of Harold Diamond Krishnaswami Alladi is professor of mathematics at the Uni- Paul Erd˝osworking in his office at the Hungarian versity of Florida. His email address is [email protected]. Academy of Sciences, Budapest. Steven Krantz is professor of mathematics at Washington University in St. Louis. His email address is [email protected]. collaborations, he influenced several generations edu. of mathematicians and molded the careers of many. DOI: http://dx.doi.org/10.1090/noti1211 He was the greatest problem proposer in history, February 2015 Notices of the AMS 121 of Ramanujan’s proof when he wrote his first paper in 1932. Erd˝oshas often jokingly said that God has a book of the most beautiful proofs of the most important theorems. His desire was to glance through this book of God (after death)! It should be pointed out that Erd˝oswas an atheist and referred to God as the Supreme Fascist. But in the case of the proofs, he admitted that God possessed this wonderful Book. Martin Aigner and Gunter Ziegler have brought out a publication [1] entitled Proofs from The Book containing the most beautiful proofs of important theorems in various branches Photo courtesy of Krishnaswami Alladi of mathematics. There are several proofs of Erd˝os An emblem of Erd˝osmade for the Erd˝os in this book, since his proofs are extremely clever, Memorial Conference in Budapest, Hungary, July elegant, and elementary. 1999. A famous problem of Erd˝oswhich is still unsolved and for which he has offered US$3,000 is and for many of his problems he offered prize the following: If fang is a sequence of increasing money depending on their difficulty. These prob- positive integers such that P 1 is divergent, then an lems have shaped the development of several areas, prove that the sequence fang contains arbitrarily and most have remained unsolved. The lectures long arithmetic progressions. An important special at the Erd˝osCentennial Conference demonstrated case of this problem when the fang is the sequence that the influence of Paul Erd˝oson the growth of of primes has been settled, and this is the celebrated mathematics remains strong. Here we have gath- Green-Tao theorem [11]. In 1936 Erd˝osand Turán ered eleven articles by leading mathematicians on conjectured that if fang has upper positive density, various aspects of Erd˝os’swork and his influence then fang contains arbitrarily long arithmetic on current research. In our article we make some progressions. Erd˝osoffered US$1,000 for the personal reflections, touch upon some of Erd˝os’s resolution of this conjecture, which was proved by work in number theory, and describe ways in which Szemerédi [20]. his legacy has been honored. Prime numbers were among Erd˝os’sfavorite Like all great geniuses, Erd˝oshad his idiosyn- topics of investigation. The prime number theorem, crasies. But even in a mathematical world used which states that the number of primes up to x to the peculiarities of its luminaries, Erd˝oswas a is asymptotic to x= log x, implies that the average most unusual phenomenon. There are hundreds gap between the nth prime pn and the next one is of Erd˝osstories that are fondly recalled at various asymptotic to log n. Two questions immediately mathematical gatherings, and this feature has a arise: (i) Can the ratio rn = (pn+1 − pn)/ log n be good sampling of such recollections. arbitrarily large? (ii) How small can this ratio Like many of the greatest mathematicians in be infinitely often? The unsolved prime twins history, Erd˝osmade his entry into the world of conjecture is the extreme solution to (ii). The research very early in his life and in a grand manner. recent sensational result of Zhang [23] on bounded His very first paper [4] was a proof of Bertrand’s gaps between primes shows that the ratio is postulate, which states that for any n ≥ 1, there O(1= log n) infinitely often (there is more about is always a prime number in the interval [n; 2n]. this in Goldston’s article in Part II of this feature. The Russian mathematician Chebychev was the See also Note Added in Proof to this article). The first to prove this, but Erd˝os’sproof utilizing the famous US$10,000 problem of Erd˝osconcerns (i). study of the prime factors of the middle binomial Westzynthius [22] in 1931 showed that the ratio rn coefficient is so elegant and clever that it is this can be arbitrarily large. Subsequently, the Scottish proof that is given in all textbooks on number mathematician Rankin [17] was able to show more theory. News of Erd˝os’sproof spread like wildfire precisely that there exists a constant c such that and was accompanied by a rhyme:“Chebychev said c: log log n: log log log log n it and I say it again, there is always a prime between r > infinitely often: n 2 n and 2n.” During many of his lectures, and in (log log log n) particular in an article entitled “Ramanujan and I” Back in 1936, Erd˝os[5] had established a similar [7], published on the occasion of the Ramanujan result but without the loglogloglogn factor in the Centennial, Erd˝oshas pointed out that there numerator. The Erd˝osUS$10,000 problem is to are similarities between his proof of Bertrand’s prove or disprove that the constant c can be postulate and Ramanujan’s, but he was not aware chosen arbitrarily large. See "Note Added in Proof" 122 Notices of the AMS Volume 62, Number 2 in this article. Ron Graham, who was the financial caretaker of Erd˝osfor many years, has a pot of money left over to give out the prizes when the problems are solved. Among Erd˝os’smany fundamental contributions it is universally agreed that the two most important are (i) his joint paper with Mark Kac [9], which ushered in the subject of probabilistic number theory, and (ii) his elementary proof of the prime number theorem along with Atle Selberg. The first proof of the prime number theorem was given toward the end of the nineteenth century of Krishnaswami Alladi simultaneously by Hadamard and de la Vallee Poussin by utilizing the properties of the zeta function ζ(s) as a function of the complex variable Courtesy s as envisioned by Riemann. The method of this Paul Erd˝osand Ernst Straus on the campus of UC proof shows that the prime number theorem is Santa Barbara during the West Coast Number Theory Conference, December 1978. equivalent to the assertion that ζ(1 + it) 6= 0 for real t. This led to the belief that any proof of the prime number theorem had to rely on the theory and n are relatively prime. The Hardy-Ramanujan of functions of a complex variable. The noted and Turán results gave a hint of probabilistic British mathematician G. H. Hardy challenged the underpinnings, but this became clear only later. In world to produce an “elementary” proof of the 1939 the great probabilist Mark Kac was giving a prime number theorem, namely, a proof that uses lecture at Princeton outlining various applications only properties of real numbers. He proclaimed of probability. He constructed a model for the study that if such an elementary proof is found, then of additive functions and conjectured that the the books would have to be rewritten, since it distribution of a wide class of additive functions would change our view of how the subject hangs about their average order and with an appropriately together. In 1949 Erd˝osand Selberg created a defined variance would be Gaussian. Erd˝os,who sensation by producing such an elementary proof. was in the audience, perked up. He realized that The elementary proof was actually found by them Kac’s conjecture could be proved using the Brun jointly by starting with a fundamental lemma of sieve, a topic in which Erd˝oswas a master. He Selberg, but for various reasons (that we shall not spoke to Kac after the lecture and they proved get into here) they had a misunderstanding and the conjecture together.