The hunt for new classes of η regulator functions

This ongoing project is part of a larger, novel research programme toward a generalised theory of regularisation. The central aim is to develop a general framework for regularisation that characterises when finite parts of divergent quantities are uniquely fixed by physical and mathematical structure. Positioned in conjunction with a number of other ongoing research projects and work packages, this project is designed specifically to support a bottom-up approach to identifying and classifying new classes of $\eta$ regulator functions, and to understand the physical principles that constrain them.

My doctoral research took initial steps toward such a framework by developing $\eta$ regularisation (missing reference) as a generalised scheme that subsumes many classical and modern summation methods (Abel, Cesàro, Ramanujan, zeta function regularisation), inspired originally by Terence Tao’s (missing reference) work on smoothed asymptotics. Defined in terms of a single axiomatic object, $\eta$ regulator functions assign values to divergent series by weighting the infinite sums, establishing many astonishing results and identities. For example, in our first paper we discussed the astonishing enhanced regulator:

$\lim_{N \to \infty} \sum_n n e^{-\frac{n}{N}} \cos\left(\frac{n}{N}\right) = -\frac{1}{12},$

and developed many interesting algorithms for finding enhanced regulators of any given order.

Ongoing work involves the search for new and interesting $\eta$ regulator classes. An important task also involves deeper study in connection with quasi-asymptotic and Stieltjes integrals, which serve as a natural generalisation of the asymptotic structures (particular in the example of the use of Schwartz functions) observed in the context of monomial and polynomial series; establishing deeper relations with zeta functions; and identifying special classes related to L-series, which are functions that encode arithmetic information into analytic objects (connecting infinite sums to prime numbers). Another important site of ongoing investigatation concerns connections with Tauberian theorems.

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