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Multifractal analysis refers to the study of the local properties of measuresand functions, and consists of two parts: the fine multifractal theory and thecoarse multifractal theory. The fine and the coarse theory are linked by a webof conjectures known collectively as the Multifractal Formalism. Very roughlyspeaking the Multifractal Formalism says that the multifractal spectrum fromfine theory equals the Legendre transform of the Renyi dimensions from thecoarse theory.  Recently {\it fine} multifractal zeta-functions, i.e. multifractalzeta-functions designed to produce detailed information about the finemultifractal theory, have been introduced and investigated. The purpose of thiswork is to complement and expand this study by introducing and investigating{\it coarse} multifractal zeta-functions, i.e. multifractal zeta-functionsdesigned to produce information about the coarse multifractal theory, and, inparticular, to establish a {\it Multifractal Fortmalism for Zeta-Functions}linking fine multifractal zeta-functions and coarse multifractal zeta-functionsvia the Legendre transform.  Several applications are given, including applications to multifractalanalysis of graph-directed self-conformal measures and multifractal analysis ofergodic Birkhoff averages of continuous functions on graph-directedself-conformal sets.

Fine and coarse multifractal zeta-functions: On the multifractal formalism for multifractal zeta-functions