  For a Borel measure and a sequence of partitions on the unit interval, we
define a multifractal spectrum based on coarse Holder regularity. Specifically,
the coarse Holder regularity values attained by a given measure and with
respect to a sequence of partitions generate a sequence of lengths (or rather,
scales) which in turn define certain Dirichlet series, called the partition
zeta functions. The abscissae of convergence of these functions define a
multifractal spectrum whose concave envelope is the (geometric) Hausdorff
multifractal spectrum which follows from a certain type of Moran construction.
We discuss at some length the important special case of self-similar measures
associated with weighted iterated function systems and, in particular, certain
multinomial measures. Moreover, our multifractal spectrum is shown to extend to
a tapestry of complex dimensions for two specific atomic measures.
