How large must 𝞓 be so that we can cover a substantial proportion of the integers below X using the binary quadratic forms x2 +dy2 with d below 𝞓? Problems involving representations by binary quadratic forms have a long history, going back to Fermat. The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao. In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition: If 𝞓 is below (log X)log 2-𝝴 then zero percent of the integers below X are represented, whereas if 𝞓 is above (log X)log 2 +𝝴 then 100 percent of the integers below X are represented.
This video is part of the Number Theory Web Seminar series.
