$\zeta-$function and heat kernel formulae

TL;DR

This paper systematically analyzes the asymptotic behavior of generalized zeta-functions and heat kernels in noncommutative geometry, clarifying their relation to Dixmier traces and strengthening existing results.

Contribution

It provides a comprehensive study of asymptotic properties and connections between zeta-functions, heat kernels, and Dixmier traces in noncommutative geometry.

Findings

Clarified the connection between zeta-functions, heat kernels, and Dixmier traces.

Strengthened and completed existing results in the literature.

Confirmed the positive answer to a question by Benameur and Fack.

Abstract

We present a systematic study of asymptotic behavior of (generalised) $\zeta-$functions and heat kernels used in noncommutative geometry and clarify their connections with Dixmier traces. We strengthen and complete a number of results from the recent literature and answer (in the affirmative) the question raised by M. Benameur and T. Fack \cite{BF}.