Linear functionals and analytic continuation problems

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

Green currents and analytic continuation

On Effective Analytic Continuation

Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis. One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic...

Analytic continuation of multiple polylogarithms

It is not hard to see that this function can be analytically continued to a multi-valued memomorhpic function on C. When n > 1 the special value Lin(1) is nothing else but the Riemann zeta value ζ(n), which has great importance in number theory. In recent years, there has been revival of interest in multi-valued classical polylogarithms and their single-valued cousins. For any positive integers...

Analytic continuation, functional equation: examples

1. L(s, χ) for even Dirichlet characters 2. L(s, χ) for odd Dirichlet characters 3. Dedekind zeta function ζ o (s) for Gaussian integers o 4. Grossencharacter L-functions L-functions for Gaussian integers We try to imitate the argument used by Riemann for proving the analytic continuation of ζ(s) and its functional equation π −s/2 Γ(s 2) ζ(s) = π −(1−s)/2 Γ(1−s 2) ζ(1 − s) from the integral rep...