Shuffle product of desingularized multiple zeta functions at any integer points

 The "multiple zeta function (MZF)" is a complex multivariable function that generalizes the Riemann zeta function. The special values of MZF at positive integer points are known as "multiple zeta values (MZV)". MZVs have captivated many mathematicians in recent years due to satisfying numerous algebraic relations such as "shuffle product relations" and "stuffle relations." In the early 2000s, the MZF was meromorphically continued to the whole complex space, and the set of all singularities of the MZF was completely determined. As a result, it became apparent that most non-positive integer points lie on the singularities of the MZF, and determining the "good" special values of the MZF at non-positive integer points became a fundamental problem in this field. Approaches such as the "renormalization method" and the "desingularization method" have been developed to address this issue. The "desingularization method" was introduced by Furusho, Komori, Matsumoto, and Tsumura in 2017, they introduced a holomorphic complex function on the whole complex space called the desingularized MZF and calculated the special values at non-positive integer points. In this talk, I will show that the special values of the desingularized MZF at all integer points satisfy the "shuffle product relations".