  The multifractal structure underlying a self-similar measure stems directly
from the weighted self-similar system (or weighted iterated function system)
which is used to construct the measure. This follows much in the way that the
dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity
dimension, is determined by the scaling ratios of the corresponding
self-similar system via Moran's theorem. The multifractal structure allows for
our definition of scaling regularity and scaling zeta functions motivated by
geometric zeta functions and, in particular, partition zeta functions. Some of
the results of this paper consolidate and partially extend the results
regarding a multifractal analysis for certain self-similar measures supported
on compact subsets of a Euclidean space based on partition zeta functions.
Specifically, scaling zeta functions generalize partition zeta functions when
the choice of the family of partitions is given by the natural family of
partitions determined by the self-similar system in question. Moreover, in
certain cases, self-similar measures can be shown to exhibit lattice or
nonlattice structure with respect to specified scaling regularity values.
Additionally, in the context provided by generalized fractal strings viewed as
measures, we define generalized self-similar strings, allowing for the
examination of many of the results presented here in a specific overarching
context and for a connection to the results regarding the corresponding complex
dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice
strings and recursive strings are defined and shown to be very closely related.
