The power of partitions and partitions into powers
演讲者
Don Zagier
时间
2025年10月22日 17:00 至 18:30
地点
A6-101
线上
Zoom 388 528 9728
(BIMSA)
摘要
The study of partitions began with Euler, who defined them and also invented generating functions to give beautiful identities and recursions for them. The generating function he found then turned up again in the 19th centuray as the expansion of the Dedekind (actually, Riemann) eta-function, which is one of the simplest and also earliest examples of a modular form. In the 20th century, partitions were taken up again by Hardy and Ramanujan, who invented the circle method to study them and gave an approximate formula with an error term that is smaller than 1/2, so that one actually gets an exact formula. I will talk about these things and also about the generalization to partitions of integers into squares or higher powers, which is much harder but involves several really nice new aspects, including a functional equation that replaces modularity in the classical case. The results depend on very subtle numerical experiments that I will also tell about, and here too there are many surprises.