Who's Who in Landau's Problems Research

At the 1912 International Congress of Mathematicians in Cambridge, Edmund Landau listed four specific problems about prime numbers as representative of what mathematics could not yet prove. Now known as Landau's problems, all four remain open more than a century later: (1) the Goldbach conjecture (every even integer greater than 2 is a sum of two primes); (2) the twin prime conjecture (infinitely many pairs of primes differing by 2); (3) Legendre's conjecture (a prime exists between every pair of consecutive perfect squares); and (4) the conjecture that infinitely many primes of the form n²+1 exist. Every one of these is unsolved.

The four problems share a mathematical core. All concern the fine-scale distribution of primes in short intervals or thin sequences, and all have been attacked by the same generation of analytic number theorists using sieve methods, the Hardy-Littlewood circle method, and exponential-sum estimates. James Maynard and Terence Tao's 2013 multidimensional sieve breakthrough, which pushed the bounded-prime-gap record to 246, also sharpened what is known about Goldbach-type representations and primes in polynomial sequences. Kaisa Matomaki and Maksym Radziwill's 2016 results on multiplicative functions in short intervals opened new routes toward Legendre's conjecture and the n²+1 problem. The community working on these problems is one community.

This site is the umbrella directory for all four problems. It catalogs the Top 100 researchers most active across the Landau-problems family, ranked from a three-source composite of arXiv preprint output, OpenAlex topical citations, and zbMATH MSC classifications. Each individual problem also has its own dedicated directory: wwigr.org (Goldbach), wwitp.org (twin primes), wwileg.org (Legendre), and wwin2p1.org (n²+1). This site captures researchers whose work spans more than one of those problems, or who contribute to the shared sieve and analytic techniques that underlie all four.