Quantum measures, arithmetic coils, and generalized fractal strings

In this work we recover the series terms of the distributional explicit formulas found in [Lap-vF1, Lap-vF2] and [Lap4] as eigenfunctions for various Hamiltonians on adelic surfaces. We interpret this result as a fundamental step in the construction of a unified physical framework in which to view the Theory of Fractal Strings and Complex Dimensions. The adelic surfaces which we construct are a direct generalization of the notion of a fractal membrane defined in [Lap4], and give a means (by way of a generalized notion of spectral partition function) of attaching the notions of prime number and integer to an arbitrary collection of complex dimensions. The construction of the adelic surfaces is made economical by our introduction of quantum measures. These measures are local complex measures which take values only in discrete quantities, and they allow us to view complex dimensions and fractal strings on an equal footing. A quantization process is defined as a map which takes a local complex measure into a quantum measure; the properties of these maps are studied and extensively generalized.