Alladi Ramakrishnan Hall

Circle Method

Manoj Verma

Post Doctoral Fellow, IMSc

Let G(k) denote the smallest natural number s such that every
sufficiently large natural number is a sum of s kth powers of non-negative
integers. In 1920s Hardy and Littlewood proved that G(k) <=
(k-2)2^(k-1)+5. Combining their ideas with an inequality proved by Hua in
1938 leads to a proof that G(k) <= 2^k+1. In this week's talk, we'll see a
proof of Hua's lemma and then finish the treatment of the minor arcs in
Waring's problem.
Afterward, we'll start with the treatment of the major arcs. I'll review
the definitions and results we covered in the last talk in case there are
any new attendees.