Sentence:  The ubiquity of the heat kernels is demonstrated in Sections 9 and 10 by constructing, respectively, the Green functions and the <WSD>Riemann zeta functions</WSD> for the Laplacians on compact Lie groups. The <WSD>Riemann zeta function</WSD> of the Laplacian on a compact Lie group is the Mellin transform of the regularized trace of the heat kernel, and we express the <WSD>Riemann zeta function</WSD> in terms of the eigenvalues of the Laplacian. Issues on the regions of convergence of the series defining the <WSD>Riemann zeta functions</WSD> are beyond the scope of this paper and hence omitted. 
Meaning of the word marked with <WSD> tags:  A usage of (a specified value of) the Riemann zeta function, such as in an equation.