Absolutely convergent cyclotomic conical zeta values

14.11.2025, 11:00  –  Campus Golm, Building 9, Room 2.22 and via Zoom
Arbeitsgruppenseminar Analysis

Bin Zhang (Chengdu) (online)

Zeta values which are discrete sums on the one dimensional cone \(]0, +\infty[\) generalise to multizeta values which are discrete sums on\( k\)-dimensional Chen cones \(0<x_k<\cdots<x_1\) with \(k\) in \(\mathbb{N}\). Going from Chen cones to general convex polyhedral cones leads to conical zeta values which in turn generalise to cyclotomic conical zeta functions when inserting a \( U(1)\)-valued character in the sum. In this talk, we show that absolutely convergent cyclotomic conical zeta values span the same space as absolutely convergent cyclotomic multiple zeta values.

For this purpose, we regularise cyclotomic conical zeta functions by means of regularised conical zeta values. We then implement subdivisions of cones combined with a rescaling resulting from symmetry properties of the roots of the unity, to reduce them to regularised cyclotomic multiple zeta values. To do so, we first reinterpret regularised conical zeta values as regularised cyclotomic matrix zeta values built from matrices. This way, we can describe transformations on the matrices involved in the cyclotomic matrix zeta values induced by subdivisions of cones applied to regularised conical zeta values. These are some of the operations on matrices we use to write regularised cyclotomic multiple zeta values as rational linear combinations of regularised cyclotomic multiple zeta values. We give a necessary and sufficient criterion for the absolute convergence of cyclotomic matrix zeta values and view absolute convergent cyclotomic multiple zeta values as limits of regularised cyclotomic matrix zeta values. In the limit we obtain that absolutely convergent cyclotomic conical zeta values can be written as  rational linear combinations of regularized cyclotomic multiple zeta values.

For more information and log in details please contact Christian Molle.