Kent State University
The question of representing a positive integer as a sum of a certain number of squares was historically a central problem in the development of modern number theory. So our starting point will be the classical problems of representability of a positive integer as the sum of two, three, or four squares (theorems of Fermat, Gauss, and Lagrange). While these problems are solved completely, they are only a small step away from unsolved, notoriously difficult problems involving sums of cubes and higher powers (Waring's problem). Using a strong form of Dirichlet's theorem for arithmetic progressions, we will give Watson's short proof of Linnik's result that every sufficiently large positive integer is the sum of at most seven cubes. Then we will finally meet the powerful Hardy-Littlewood method, through Hua's elegant proof that at most 2k+1 summands suffice to represent every sufficiently large positive integer as the sum of kth powers. We will conclude with a survey of the state of knowledge in Waring's problem, Goldbach's Conjecture (that every even number larger than 2 is the sum of two primes), and the theory of partitions.
Afternoon (break-out) sessions will include problems meant to fill in proofs from the lectures, as well as theorems and problems from the theory of partitions. For example, we might attempt to prove, in digestible steps, the Hardy-Ramanujan-Rademacher formula for the number p(n) of partitions of n.
Reading list: The first volume of Mel Nathansons's two Springer-Verlag books, "Additive Number Theory", subtitled "The Classical Bases", contains much of the material we wish to cover, and are easily obtained.
Books abound on partitions: Apostol's two Springer-Verlag books on Analytic Number Theory are both in print and contain both elementary and advanced theorems.
I recommend reading the original paper of Hardy and Ramanujan which was the genesis of the circle method: "Asymptotic formulae in combinatory analysis" Proc. London Math. Soc. (2) 17 (1918), 75-115.
Parts of this and other articles will be required reading during the week.