Graduate Student Number Theory Learning Seminar
Waring's Problem and the Circle Method
Nathan Mehlhop (Rutgers University)
Date & time: Tuesday, 06 April 2021 at 4:00PM - 5:00PM
Abstract: In 1770, Lagrange showed that every natural number can be written as the sum of four perfect squares. Is there always a finite number g(k) of terms needed when we instead consider positive kth powers? This is Waring's problem, and it was proven in 1909 by Hilbert, but the method gave no useful numerical bounds. In 1919, Hardy and Littlewood introduced the circle method and gave asymptotics for the number of representations of N as the sum of s many positive kth powers. This method is widely applicable to questions of additive number theory. I will introduce the circle method in the context of Waring's problem.