Algebra Seminar

Speaker: Sheldon Joyner (Western)

"Zeta functions as iterated integrals"

Time: 13:30

Room: MC 108

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.

The Riemann zeta function, (along with Dirichlet L-functions and the polyzeta functions), may be expressed as a complex iterated integral which can be regarded as a transform of a certain rational function (in keeping with the general philosophy that zeta functions should be rational). The eventual hope is that these expressions will endow such functions with new geometric meaning, mimicking the role the polyzeta values play as periods - which is owing to certain (usual) iterated integral expressions.

On the other hand, the expected irrationality of the residue of the pole of the Dedekind zeta function \zeta_K(s) (for K <> Q) at s=1 is an obstruction to the function arising from the complex iterated integral expression for \zeta_K(s) being algebraic (let alone rational). This in turn happens to produce an obstruction to the existence of a proof of the functional equation for the Dedekind zeta function along the lines of Riemann's contour integration proof of the functional equation for \zeta(s).

As a final application, we shall mention an elegant reformulation of a result of Gel'fand and Shilov in the theory of distributions, which gives a way of thinking of complex iterated derivatives.