Research
My research in number theory centers
aroung the study of the multiple L-values of Euler-Zagier type. The
prototypical examples are the multiple zeta values (or MZV). This
is a very young subject; the modern study of these numbers
began only after they appeared in some quantum physics calculations
in the 1980's. The MZV have since been linked with cohomology
of motives, knot theory, and quantum physics. The MZV have an
extensive and interesting algebraic structure, which has been studied
by Hoffman, Goncharov, and others. By way of analogy with the
classical situation, one inserts Dirichlet characters to obtain
multiple L-values (MLV). The MLV have algebraic structures
precisely analogous to those of the MZV.
As in
the classical situation, one can consider these sums as functions of
complex variables, and in several cases, the existence of a
meromorphic continuation has been shown. However, functional
equations of the type satisfied by the classical L-functions have not
arisen.
My approach to studying the MLV is
mostly analytic. My long-range research goal is to investigate
the possibility of a general multiple L-function theory (perhaps
analogous to that of the classical L-functions). One
hopes also for connections with modularity and transcendence
(concerning, e.g., the Riemann zeta values).
In graduate school, I studied the MZV and MLV under Fernando
Rodriguez-Villegas at UT-Austin. In my thesis, I used methods
of J. Borwein and R. Girgensohn to generalize an evaluation theorem
for the double zeta values to the double L-values, and investigated
an analogous property for the triple L-values. The essential
technique involves partial fractions. We also generalized the
method of Crandall for fast computation of MZV to the MLV.
Numerical computation plays an important role in discovering these
evaluations - below is a link to a file containing PARI commands for
the series we used.
In [3], I give a
complex analytic proof of a piece of my thesis result. My
approach was inspired by work of Tsumura on the MZV and the
Mordell-Tornheim zeta values. The method uses the monodromies
of new polylog-type functions I define, and the parities of
generating functions of generalized Bernoulli numbers. Some
interesting formulas are produced which hold for quite general
characters, of which an example is given.
Papers
[1] Evaluations of Multiple L-values,
Ph.D. thesis, UT-Austin, 2002. ps
dvi pdf
[2]
Evaluations of Double L-values, J. Num. Th., vol. 105/2,
2004. ps
dvi pdf
[3]
Evaluations of a Class of Double L-values, to appear in Proc.
Amer. Math. Soc. dvi
pdf