Abstract:

In 1978, Apery proved the irrationality of the Riemann zeta value ζ(3) by utilizing a fast converging sequence of rational approximations. However, the details of his proof remained difficult to comprehend. Since then, many mathematicians have attempted to either elucidate Apery's approach or seek out new proofs altogether.
Inspired by computer algorithms to find such approximations, an idea about a hidden mathematical structure behind the proof was beginning to formalize. This structure, which we call the conservative matrix field, not only clarifies some of Apery's proof but also offers a framework for Apery-like irrationality proofs and relates them to the broader themes of number theory and dynamics. In this talk, we will explore the properties of the conservative matrix field and discuss how it can be hopefully applied to prove irrationality for other mathematical constants.
This research was done as part of the work in the Ramanujan Machine group.