Olivetti Club

Itamar Oliveira Cornell University
The circle method

Tuesday, March 2, 2021 - 4:30pm

Let $P(n)$ be the number of ways of writing $n$ as a sum of positive integers without counting reorderings (e.g. $P(4)=5$). Is there a simple formula for $P(n)$? If not, how does it grow as $n \rightarrow \infty$? Another problem: what is the smallest natural number $s$ for which any $n$ can be written as a sum of perfect $k$-th powers? Another one: is it true that any even number bigger than $2$ is the sum of two primes? I'm sure this rings a bell.

Hardy and Ramanujan successfully found asymptotics for $P(n)$ by working on the problem in a framework nowadays called The circle method. In this talk, we will start from basic ideas in combinatorics and use contour integration to describe the main steps of this mechanism. We will also see how it can be used to deal with problems such as the ones in the previous paragraph. If there is still someone in the audience after all this, I hope to show a link with Harmonic Analysis and why an analyst would care about it.