This article continues an extensive analysis of spectral functions such as for both general and explicitly known spectra {λm}. In physical applications (which in quantum field theory are numerous) the spectral functions are mode sums. Our main analytic tool is the ζ-function resummation method which expresses f(s|x) in powers of x and perhaps other simple functions of x. Here the general spectrum will be replaced by its asymptotic form λm = (const) mα with α > 0 (Weyl's theorem). This preserves certain global features of the general spectrum problem but enables one to work entirely in terms of known functions. This simplified problem will be fully analysed and certain aspects of it (in particular the continuum limit) studied for the first time. Several mode-sum calculations illustrate physical application of the method.