We derive a natural generalization of the Krein and Neidhardt trace formulas for tuples of commuting unitaries via a multiplicative path, allowing perturbations to be in a family of Lorentz ideals of bounded linear operators.

The authors would like to heartily thank the anonymous referee for a careful reading of the manuscript and several helpful suggestions. They also gratefully acknowledge useful conversations with Chandan Pradhan. A. Chattopadhyay is supported by the Core Research Grant (CRG), File No: CRG/2023/004826, from SERB. S. Giri is grateful to the Ministry of Education, Government of India, for financial support through the Prime Minister's Research Fellowship (PMRF-ID: 1902164) grant.