The age of 40 is considered special in mathematics because it’s an ad-hoc criterion for deciding whether a mathematician is young or old. This idea has been well established by the under-40 rule for Fields Medal, based on Fields‘ desire that:

while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others

Though it must be noted that this criterion doesn’t claim that after 40 mathematicians are not productive (example: Yitang Zhang). So I wanted to write a bit about the under 40 leading number theorists which I am aware of (in order of decreasing age):

**Sophie Morel**: The area of mathematics in which Morel has already made contributions is called the*Langlands program*, initiated by Robert Langlands. Langlands brought together two fields,*number theory*and*representation theory.*Morel has made significant advances in building that bridge. “It’s an extremely exciting area of mathematics,” Gross says, “and it requires a vast background of knowledge because you have to know both subjects plus*algebraic geometry*.” [source]**Melanie Wood:**Profiled at age 17 as “The Girl Who Loved Math” by Discover magazine, Wood has a prodigious list of successes. The main focus of her research is in*number theory*and*algebraic geometry*, but it also involves work in probability, additive combinatorics, random groups, and algebraic topology. [source1, source2]**James Maynard:**James is primarily interested in classical*number theory*, in particular, the distribution of prime numbers. His research focuses on using tools from analytic number theory, particularly*sieve methods*, to study primes. He has established the sensational result that the gap between two consecutive primes is no more than 600 infinitely often. [source1, source2]**Peter Scholze**: Scholze began doing research in the field of*arithmetic geometry,*which uses geometric tools to understand whole-number solutions to polynomial equations that involve only numbers, variables and exponents. Scholze’s key innovation — a class of fractal structures he calls*perfectoid spaces*— is only a few years old, but it already has far-reaching ramifications in the field of arithmetic geometry, where number theory and geometry come together. Scholze’s work has a prescient quality, Weinstein said. “He can see the developments before they even begin.” [source]