# James Maynard (mathematician)

James Maynard
Born10 June 1987 (age 33)
Chelmsford, England[1]
NationalityBritish
Alma materUniversity of Cambridge
University of Oxford
Known forWork on prime gaps
Awards
Scientific career
FieldsMathematics
InstitutionsUniversity of Montreal
University of Oxford

James Maynard (born 10 June 1987) is a British mathematician best known for his work on prime gaps.[1] After completing his bachelor's and master's degrees at University of Cambridge in 2009, Maynard obtained his D.Phil. from University of Oxford at Balliol College in 2013 under the supervision of Roger Heath-Brown.[2][1] For the 2013–2014 year, Maynard was a CRM-ISM postdoctoral researcher at the University of Montreal.[3] In 2017, he was appointed Research Professor at Oxford.[4]

## Work

In November 2013, Maynard gave a different proof of Yitang Zhang's theorem[5] that there are bounded gaps between primes, and resolved a longstanding conjecture by showing that for any ${\displaystyle m}$ there are infinitely many intervals of bounded length containing ${\displaystyle m}$ prime numbers.[6] This work can be seen as progress on the Hardy–Littlewood ${\displaystyle m}$-tuples conjecture as it establishes that "a positive proportion of admissible ${\displaystyle m}$-tuples satisfy the prime ${\displaystyle m}$-tuples conjecture for every ${\displaystyle m}$."[7] Maynard's approach yielded the upper bound

${\displaystyle \liminf _{n\to \infty }\left(p_{n+1}-p_{n}\right)\leq 600,}$

which improved significantly upon the best existing bounds due to the Polymath8 project.[8] (In other words, he showed that there are infinitely many prime gaps with size of at most 600.) Subsequently, Polymath8b was created,[9] whose collaborative efforts have reduced the gap size to 246.[8]

On 14 April 2014, one year after Zhang's announcement, according to the Polymath project wiki, N had been reduced to 246.[8] Further, assuming the Elliott–Halberstam conjecture and, separately, its generalised form, the Polymath project wiki states that N has been reduced to 12 and 6, respectively.[8]

In August 2014, Maynard (independently of Ford, Green, Konyagin and Tao) resolved a longstanding conjecture of Erdős on large gaps between primes,[10] and received the largest Erdős prize (\$10,000) ever offered.[11]

In 2014, he was awarded the SASTRA Ramanujan Prize.[1][12] In 2015, he was awarded a Whitehead prize and in 2016 an EMS Prize.

In 2016, he showed that, for any given decimal digit, there are infinitely many prime numbers that do not have that digit in their decimal expansion.[13]

In 2019, together with Dimitris Koukoulopoulos, he proved the Duffin–Schaeffer conjecture.[14]

## References

1. ^ a b c d Alladi, Krishnaswami. "James Maynard to Receive 2014 SASTRA Ramanujan Prize" (PDF). qseries.org. Retrieved 13 April 2017.
2. ^ a b
3. ^ "Dr James Maynard". Magdalen College, Oxford. Archived from the original on 20 May 2018. Retrieved 17 April 2014.
4. ^
5. ^ Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. Princeton University and the Institute for Advanced Study. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. Retrieved 16 August 2013.
6. ^ Klarreich, Erica (19 November 2013). "Together and Alone, Closing the Prime Gap". Quanta Magazine). Archived from the original on 5 December 2019. Retrieved 5 December 2019.
7. ^ Maynard, James (20 November 2013). "Small Gaps Between Primes". arXiv:1311.4600 [math.NT].
8. ^ a b c d "Bounded gaps between primes". Polymath Project. Retrieved 21 July 2013.
9. ^ Tao, Terence (19 November 2013). "Polymath8b: Bounded intervals with many primes, after Maynard".
10. ^ Maynard, James (21 August 2014). "Large gaps between primes". arXiv:1408.5110 [math.NT].
11. ^ Klarreich, Erica (10 December 2014). "Prime Gap Grows After Decades-Long Lull". Quanta Magazine. Retrieved 10 December 2014.
12. ^ Alladi, Krishnaswami (December 2014), "Maynard Awarded 2014 SASTRA Ramanujan Prize" (PDF), Mathematics People, Notices of the AMS, 61 (11): 1361, ISSN 1088-9477.
13. ^ Maynard, J.: Invent. math. (2019) 217: 127. https://doi.org/10.1007/s00222-019-00865-6
14. ^ Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593. Bibcode:2019arXiv190704593K. Cite journal requires |journal= (help)