On a generalization of Chen's iterated integrals

TL;DR

This paper extends Chen's iterated integrals to complex values, exploring their algebraic properties and applications such as zeta functions, distribution theory, and polylogarithm monodromy, revealing new insights and obstructions in these areas.

Contribution

It introduces a generalized theory of iterated integrals for complex parameters, establishing their algebraic properties and demonstrating diverse applications in number theory and analysis.

Findings

Expressed zeta functions as complex iterated integrals

Reformulated distribution theory results using complex iterated derivatives

Provided a topological proof of polylogarithm monodromy

Abstract

Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive and a (non-classical) multiplicative iterative property, in addition to a comultiplication formula. This theory is developed in the first part of the paper, after which various applications are discussed, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the existence of a contour integration proof of the functional equation for the Dedekind zeta function emerges); an elegant reformulation of a result of Gel'fand and Shilov in the theory of distributions which gives a way of thinking about complex iterated derivatives; and a direct topological proof of the monodromy of polylogarithms.