2014-12-94

Newsconjecture of Elliott and Halberstam on the distributionof primes in arithmetic progression holds then the gapcould be made as small as 16 infinitely often.Two years ago, Y. Zhang stunned the world by showingthat infinitely often the smallest gap is no more than70 million! This was the first time a bounded gap was establishedunconditionally. For this seminal work on thesmall gap problem, Goldston, Pintz, Yildirim and Zhangreceived the 2014 Cole Prize of the American MathematicalSociety.Zhang’s method was quite complex. He had to circumventthe use of a hypothesis that went beyond the Bombieri–Vinogradovtheorem. Terence Tao, by leading thePolymath project, reduced Zhang’s bound to 4680. Butwithin a few months of Zhang’s achievement, Maynardtook the world by storm by establishing, in a simpler andmore elegant fashion, that the gap between the primes isno more than 600 infinitely often! This sensational paperon “Small gaps between primes” will soon appear in theAnnals of Mathematics. Actually, in this paper, Maynardestablishes a number of other deep results. For example,he shows that for any given integer m, the gap betweenthe n + m-th prime and the n-th prime is bounded by aprescribed function of m. Very recently, Maynard hasjoined the Polymath project and now the gap betweenconsecutive primes has been shown to be no more than246 infinitely often, by adapting the method in Maynard’spaper in the Annals of Mathematics. This is to appear inthe journal Research in the Mathematical Sciences.Within the last month, Maynard has announced asolution of the famous $10,000 problem of Paul Erdősconcerning large gaps between primes. This was also simultaneouslyannounced by Kevin Ford, Ben Green,Sergei Konyagin and Terence Tao but Maynard’s methodsare different and simpler. In 1938, Robert Rankinestablished a lower bound for infinitely many large gaps,which remained for many years the best result on thelarge gap problem. The great Hungarian mathematicianPaul Erdős asked whether the implicit constant inRankin’s lower bound could be made arbitrarily largeand offered $10,000 to settle this question either in theaffirmative or in the negative. In the last 50 years, therehave been improvements made by various leading researcherson the value of the implicit contant in Rankin’sbound – by Rankin himself in 1962, by Helmut Maier andCarl Pomerance in 1990 and by Janos Pintz in 1997 – butthe Erdős problem still remained unsolved. Maynard hasnow shown that the implicit constant in Rankin’s lowerbound could be made arbitrarily large. So in settling thisnotoriously difficult problem, an impasse of several decadeshas been broken. Another fundamental recent workof Maynard is a collaboration with William Banks andTristan Freiburg on the limit points of the values of theratio of the gap between consecutive primes and the averagegap. Results on the small gaps problem imply that0 is a limit point and the work on the large gap problemshows that ∞ is a limit point. No other limit point isknown. In this joint paper it is shown that if 50 randomreal numbers are given then one at least of the 50 consecutivegaps between them and 0 will occur as a limitpoint. Thus, in the last three years, Maynard has obtainedseveral spectacular results in the theory of primes bymethods which will have far reaching implications.James Maynard was born in Chelmsford, England,on 10 June 1987. He obtained his BA and Master’s inMathematics from Cambridge University in 2009. Hethen joined Balliol College, Oxford University, wherehe received his Doctorate in Philosophy in 2013. During2013–14 he was a post-doctoral fellow at the Universityof Montreal, Canada. Within a period of three years,Maynard has startled the mathematical world with deepresults in rapid succession. The SASTRA RamanujanPrize will be his first major award in recognition of hismany fundamental results in prime number theory.The 2014 SASTRA Ramanujan Prize Committeeconsisted of Professors Krishnaswami Alladi – Chair(University of Florida), Roger Heath-Brown (OxfordUniversity), Winnie Li (Pennsylvania State University),David Masser (University of Basel), Barry Mazur (HarvardUniversity), Peter Paule (Johannes Kepler Universityof Linz) and Michael Rapoport (University ofBonn).Previous winners of the prize are Manjul Bhargavaand Kannan Soundararajan in 2005 (two full prizes), TerenceTao in 2006, Ben Green in 2007, Akshay Venkateshin 2008, Kathrin Bringmann in 2009, Wei Zhang in 2010,Roman Holowinsky in 2011, Zhiwei Yun in 2012 and PeterScholze in 2013. The award of the 2014 SASTRA RamanujanPrize to James Maynard is a fitting recognitionin the tenth year of this prize and is in keeping with thetradition of recognising groundbreaking work by youngmathematicians.Krishnaswami Alladi is a professor ofmathematics at the University of Florida,Gainesville, where he was DepartmentChairman during 1998–2008. Hereceived his PhD from UCLA in 1978.His area of research is number theory.He is the Founder and Editor-in-Chiefof The Ramanujan Journal publishedby Springer. He helped create the SAS-TRA Ramanujan Prize given to veryyoung mathematicians and has chaired the prize committeesince its inception in 2005.EMS Newsletter December 2014 11