論文の公開元へRenormalized values and desingularized values of the multiple zeta function
概要
It is known that there are infinitely many singularities of multiple zeta functions and almost all negative integer points are located in their singularities. This causes an indeterminacy of the special values there. It is a fundamental problem to give a nice definition of special values of multiple zeta functions at non-positive integers. As one approach, Guo and Zhang ([GZ]) used the renormalization procedure which arises from the Hopf algebraic approach to perturbative quantum field theory by Connes and Kreimer to introduce the renormalized values as a special values of multiple zeta functions. Other type of renormalized values were introduced by Manchon and Paycha ([MP]) and Ebrahimi-Fard, Manchon and Singer ([EMS16], [EMS17]).
While, Furusho, Komori, Matsumoto and Tsumura ([FKMT17a]) proposed the desingularization method to resolve all singularities of multiple zeta functions, and by using this method, they introduced the desingularized multiple zeta functions, which can be analytically continued to the whole space as entire functions. The desingularized values are defined to be the special values of desingularized multiple zeta functions at integer points. They gave explicit formulae of these special values in terms of Bernoulli numbers. The aim of this thesis is to give a concrete relationships among desingularized values and various renormalized values.
In Chapter 1, We recall the definition of multiple zeta functions and the desingularized multiple zeta functions introduced by Furusho, Komori, Matsumoto and Tsumura, and we explain various properties of the desingularized multiple zeta functions. In Chapter 2, we consider the renormalized values introduced by Ebrahimi-Fard, Manchon and Singer, and the relationship between desingularized values in [FKMT17a] and renormalized values in [EMS17]. In §2.1, we review the definition of the Hopf algebra H0, which is used to define these renormalized values in §2.2. In §2.3, we give explicit formulae for the coproduct ∆0 of the Hopf algebra H0, which are used to prove the recurrence formulae among renormalized values in §2.4. In §2.5, by using these recurrence formulae, we prove an equivalence between desingularized values and renormalized ones and by using this equivalence, we give the explicit formula of the renormalized values in terms of Bernoulli numbers. In Chapter 3, we consider functional relations of desingularized multiple zeta functions. In §3.1, we prove the product formulae of desingularized multiple zeta functions at nonpositive integer points. In §3.2 and §3.3, we prove functional relations of desingularized multiple zeta functions as a generalization of that product formulae at non-positive integer points in two different ways. Chapter 4 is on a problem posed by Singer which is on a comparison between the renormalized values of shuffle type and of harmonic type. We settle the problem by giving a universal presentation of the renormalized values introduced by Ebrahimi-Fard, Manchon and Singer as finite linear combinations of any renormalized values of harmonic type.
This doctor thesis is based on three papers [Ko19], [Ko20a], [Ko20b], and current research. In precise, the section Chapter 2 is based on [Ko19], and the sections §3.1 and §3.2 are based on [Ko20a], and the section §3.3 is based on [Ko20b], and the section Chapter 4 is the ongoing research.
