Since the 1990's, certain real numbers, called MZVs (multiple zeta
values), have been actively studied due to their connections with
various areas of mathematics.
One noteworthy property is that they satisfy a huge number of algebraic
relations.
It was proved by Hoffman that some relations can be understood in the
context of Hopf algebra.
From an analytic point of view, MZVs can be regarded as the values of
the MZF (multiple zeta function) at positive integer points.
In the simplest case of MZFs, namely the classical Riemann zeta
function,
it is well known that Riemann zeta function has the analytic
continuation to the whole complex plane and that the values at non
positive integer points can be expressed by the Seki-Bernoulli numbers.
Similarly, it is known that MZFs can be meromorphically continued to the
whole complex space,
and their values and algebraic properties at non-positive integer points
have been studied by many mathematicians.
However, the values and algebraic properties of MZFs at all integer
points remain less well understood.
In this talk, I will discuss methods for computing the values of MZFs at
all integer points and explore algebraic properties using several
techniques from Hopf algebras.
