2014-12-94

NewsJames Maynard to Receive 2014SASTRA Ramanujan PrizeKrishnaswami Alladi (University of Florida, Gainsville, USA), Chair SASTRA Ramanujan Prize CommitteeThe 2014 SASTRA Ramanujan Prize will be awarded toDr James Maynard of Oxford University, England, andthe University of Montreal, Canada.The SASTRA Ramanujan Prize was established in2005 and is awarded annually for outstanding contributionsby young mathematicians to areas influenced bythe genius Srinivasa Ramanujan. The age limit for theprize has been set at 32 because Ramanujan achieved somuch in his brief life of 32 years. The prize will be awardedon 21–22 December at the International Conferenceon Number Theory at SASTRA University in Kumbakonam(Ramanujan’s hometown), where the prize hasbeen given annually.Dr Maynard has made spectacular contributions tonumber theory, especially on some famous problems onprime numbers. The theory of primes is an area wherequestions which are simple to state can be very difficultto answer. A supreme example is the celebrated “primetwins conjecture”, which states that there are infinitelymany prime pairs that differ by 2. In the last 12 months,Dr Maynard has obtained the strongest result towardsthis centuries old conjecture by proving that the gap betweenconsecutive primes is no more than 600 infinitelyoften. Not only did he significantly improve upon theearlier groundbreaking work of Goldston, Pintz, Yildirimand Zhang but he achieved this with ingenious methodsthat are simpler than those used by others. Maynard’s remarkablesuccess on the “small gap problem” on primesis built upon the ideas and results he developed in his doctoralwork a few years ago on some other famous problemson primes such as the Brun–Titchmarsh inequality.Maynard’s results and methods have led to a resurgenceof activity worldwide in prime number theory.In his doctoral thesis of 2013 at Oxford University,written under the direction of Professor Roger Heath-Brown, Maynard obtained a number of deep results onsome fundamental problems. One of the intriguing andimportant questions on prime numbers is how uniformlythey are distributed in various arithmetic progressionsof integers up to a given magnitude. There is an heuristicestimate for the number of such primes in arithmeticprogressions and one usually gets a bound of the order ofmagnitude given by the heuristic. This Brun–Titchmarshproblem becomes difficult when the modulus or gap betweenthe members of the progression becomes verylarge and results are weaker compared to the conjecturedsize. Earlier researchers had relied on a certain unprovenhypothesis concerning the Siegel zeros of L-functions inorder to treat these large moduli. But Maynard showedin his thesis (without appeal to any hypothesis) that thenumber of primes in arithmetic progressions is boundedby the suspected heuristic size for arithmetic progressionswhose moduli do not exceed the eighth root of thelargest member of the progression. This important paperappeared in Acta Arithmetica in 2013.A generalisation of the prime twins conjecture is theprime k-tuples conjecture, which states that an admissiblecollection of k linear functions will simultaneouslytake k prime values infinitely often in values of theargument. In the last 100 years, several partial resultstoward the k-tuples conjecture have been obtained eitherby replacing prime values of some of these linearfunctions by “almost primes” (which are integers with abounded number of prime factors) or by bounding thetotal number of prime factors in the product of these linearfunctions. Another major achievement in his doctoralthesis is his work on “Almost-prime k-tuples” (which waspublished in Mathematika in 2014) in which he obtainsbounds for the number of prime factors in the product ofthese admissible linear functions. These bounds are superiorto bounds obtained by earlier researchers, exceptin the case of the product of three linear functions, wherehis result was just as strong as the 1972 result of Porter,who had confirmed that the triple product will have nomore than eight prime factors infinitely often. But in aseparate paper that appeared in the Proceedings of theCambridge Philosophical Society in 2013, Maynard brokethe impasse by improving on Porter’s result and showingthat a product of three admissible linear functions willhave no more than seven prime factors infinitely often.In establishing these fundamental results, Maynard introduceda number of new methods and techniques thatenabled him to achieve a sensational result on the smallgaps problem on primes within a year of completinghis DPhil – when he was a post-doctoral fellow at theUniversity of Montreal, Canada, with Professor AndrewGranville as his mentor.The prime number theorem implies that the averagegap between the n-th prime and the next prime is asymptoticto log n. Two questions arise immediately:(i) How small can this gap be infinitely often (the smallgap problem)? and(ii) How large can this gap be infinitely often (the largegap problem)? The prime twins conjecture says thatthe gap is two infinitely often.It was a sensation a few years ago when Goldston, Pintzand Yildirim (GPY) showed that over a sequence of integersn that tend to infinity, the gap between the n-thprime and the next can be made arbitrarily smaller thanthe average gap. It was shown by GPY that if a certain10 EMS Newsletter December 2014