We study the essential singularities of geometric zeta functions
ζL, associated with bounded fractal strings L. For
any three prescribed real numbers D∞, D1 and D in [0,1], such
that D∞<D1≤D, we construct a bounded fractal string L
such that Dpar(ζL)=D∞, Dmer(ζL)=D1 and D(ζL)=D. Here,
D(ζL) is the abscissa of absolute convergence of
ζL, Dmer(ζL) is the abscissa of
meromorphic continuation of ζL, while Dpar(ζL) is the infimum of all positive real numbers α
such that ζL is holomorphic in the open right half-plane
{Res>α}, except for possible isolated singularities in this
half-plane. Defining L as the disjoint union of a sequence of
suitable generalized Cantor strings, we show that the set of accumulation
points of the set S∞ of essential singularities of ζL, contained in the open right half-plane {Res>D∞},
coincides with the vertical line {Res=D∞}. We extend this
construction to the case of distance zeta functions ζA of compact sets
A in RN, for any positive integer N.Comment: Theorem 3.2 (b) was wrong in the previous version, so we have decided
to omit it and pursue this issue at some future time. Part (b) of Theorem
3.2. was not used anywhere else in the paper. Theorem 3.2. is now called
Proposition 3.2. on page 12. Corrected minor typos and added new references
To appear in: Pure and Applied Functional Analysis; issue 5 of volume 5
(2020