I approach this issue constructively. I'll give you my example, and you can point me to other examples of physical interpretation of arithmetic functions. My example in the attached paper "On the winding of a sphere" in which are the conclusion:
"Thus, our abstract mathematical constructions have some similarities with the mathematical formalism of quantum mechanics, and therefore the questions of the substantiation of quantum mechanics could possibly get an answer within the framework of the mathematical formalism developed here. At least, bearing in mind that the Jacobi theta function $\theta(z, \tau)$ satisfies the Schrodinger equation with complex variables, we could interpret it as ``quantum'' oscillations of the mathematical pendulum of the sphere winding, and Hurwitz zeta function satisfying the generalized Schrodinger equation, interpreted as a function of the oscillations of a mathematical pendulum with an evolving (for example, decaying) complex angle of deflection of the winding. On the other hand, the metaphysical method of random walks around winding a sphere that we have developed will probably find application in the substantiation of the Hilbert-Polya conjecture on the connection of nontrivial zeros of the Riemann zeta function with the eigenvalues of a certain differential operator."
and the abstract:
"First, we bring the reader to one remarkable result of the action of a modular group on a sphere, proving that of all closed torus windings wound around a sphere, single-wound windings that are indexed by a set of primes stand out. Further, we show that the rotation of the torus windings on a sphere, together with the measurement of the complex value of the angular coordinates of a discrete set of their points, gives us all the necessary data for the formation of the Riemann zeta function. Then, considering the dynamics of the windings, we notice that in the problem of random walk along the broken lines of the winding of a sphere, the concept of complex probability amplitude arises quite naturally, and the dynamics of the probability amplitude of the stray particle obeys a differential equation generalizing the Schrodinger equation."
I will only add that, perhaps, we can find a technological application of this equation.
Indeed, if the generalized Schrödinger equation works in nature, then the interaction time should be included as an additional factor in the nuclear reaction. In other words, due to the exponentially time-dependent coefficient of the generalized Schrödinger equation, the longer the nuclei come closer, the higher the probability of a nuclear reaction.
Preprint On the winding of a sphere
Metaphysical interpretation of arithmetic functions is not strange. You can interpret mathematical formulations in thousand of different mediums. Mathematics is nothing but approximate measurable representation or coding of the physical as well as observable unseen world and even universe. For example, look at purpose/application of Ontology i.e., to to arrange best fitting categories or classes in special order., It is equivalent to optimization or classification or pattern recognition. In particular Metaphysics and Mathematics have very close relationship. Both are about studding/investigating the realities about things, however., Metaphysics discusses Philosophical concepts, whilst Mathematics analyses the real visual and computational concepts.
Muhammad Ali, I am glad that I met a like-minded person here, but I would like to note that mathematical models describe reality (nature) in some cases, and fantasy (invented nature) in others. My goal is to describe moving matter.
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