Zeta functions as iterated integrals

Sheldon Joyner, Purdue University

Abstract

Values of the polyzeta functions at tuples of positive integers arise as periods relating the rational structures underlying the topological and algebraic realizations of the motivic fundamental group of [special characters omitted]1\{0, 1,∞}. A first step towards conferring similar arithmetic meaning to more general values of the polyzeta functions is taken in this thesis, in which it is shown that an appropriate notion of iterated integral expression interpolating that of CHEN may be defined and shown to satisfy a suitable iterative property. The new formalism provides intrinsic motivation for the classical M ELLIN transform and coincides with the fractional integral of L IOUSVILLE and RIEMANN in the simplest case, but admits generalization to iteration of any holomorphic 1-form on a given complex manifold subject to some iterative property. Also, a coproduct formula extending that on usual iterated integrals exists. The naturality of the complex iterated integral perspective is evidenced by the new light it sheds on a theorem of GEL’FAND and SHILOV in the theory of distributions. A further application is the association of power series to DIRICHLET L-functions and DEDEKIND zeta functions, by means of which we show that the irrationality of the residue of the pole of the latter at s = 1 is an obstruction to the existence of an analytic continuation along the lines of the analytic continuation of the RIEMANN zeta function by means of a contour integral.

Degree

Ph.D.

Advisors

Kim, Purdue University.

Subject Area

Mathematics

COinS