Sheldon Joyner
In progress
Preprints
With a view towards establishing an analogue of Hecke's wonderful theorem giving (modulo certain technical details) the equivalence of the analytic continuation and functional equation of certain L-functions with modularity of associated forms, I showed in this paper that certain sections of the universal prounipotent bundle with connection on the thrice-punctured sphere may be endowed with an action of the modular group,
(by means of twisted parallel transport along paths in an explicit path space realization of PSL(2,Z)). For a distinguished section of the bundle, namely the polylogarithm generating function, a 1-cocycle reminiscent of a classical automorphy factor arises, and leads to the definition of the quasi-modularity of the title. It turns out that this quasi-modularity encodes the data used to prove the analytic continuation and functional equation of the Riemann zeta function in my previous work.
This paper presents some new proofs of the analytic continuation and functional equation of Riemann's zeta function - in fact, one proof for each integer other than 1. My goal here was to explore the geometry underlying Riemann's contour integral proof of these facts, and I showed that data coming from the KZ equation (which is a prounipotent connection on a suitable bundle over P^1\{0,1,\infty} satisfying a universality property) informs the proofs at positive integer parameters; while those at negative integer parameters rely on a formula with an elementary expression but a non-elementary proof, which encodes monodromy information.
The shuffle algebra of multiple zeta values may be described using the polylogarithm functions. In this paper, I focus on the stuffle algebra from the analgous perspective provided by the multiple Hurwitz zeta functions. Using the generating function H(z) of such functions, a difference equation analogue of the Knizhnik-Zamolodchikov equation satisfied by the generating series of polylogarithms is exhibited. Modelled on this algebraic KZ equation, a difference equation analogue of the usual notion of connection is then defined, for which H(z) is a flat section. By solving certain difference equations, the counterparts of the multiple Hurwitz polyzeta functions at tuples of negative integers are also determined. These are generalized multiple Bernoulli polynomials, a generating function for which is shown to satisfy a similar difference equation to that for H(z). Certain values of these polynomials give regularizations of multiple zeta values, which are shown to agree with the regularizations given in work of S. Akiyama and collaborators. Finally, the algebraic independence of the Hurwitz polyzeta functions is proven, to contrast with the fact that the multiple Bernoulli polynomials have rational coefficients.
Journal articles
This paper details the results of the thesis (see below) along with some refinements and further results.
ThesesPlease email me for a copy of either thesis
In my thesis, I found a way to interpolate certain Chen iterated integrals to allow (in some sense) for integrating against some complex power of a holomorphic 1-form. This facilitates the expression of arithmetic L-functions as iterated integrals, and allowed me to show that irrationality of the residue of the Dedekind zeta function at s=1 is an obstruction to the existence of a contour integral proof (modelled after that of Riemann in the case of his zeta function) of the analytic continuation and functional equation. Besides developing a multiple version of the definition, and a generalization to arbitrary complex manifolds, I showed that a coproduct formula holds, by means of which a direct computational proof was given of the monodromy of the classical polylogarithm functions. These results are expounded in the J. Number Theory paper.