The ubiquitous ζ-function and some of its 'usual' and 'unusual' meromorphic properties

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated with conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.