We provide exact expressions for the volume of polyhedral neighborhoods for the sequence of prefractal graphs which converge to the Weierstrass Curve, called Weierstrass Iterated Fractal Drums (in short, Weierstrass IFDs), associated with a suitable (and geometrically meaningful) sequence of values of a parameter \(\epsilon \) tending to zero, also known as the cohomology infinitesimal, due to its connections with fractal cohomology, as developed in an earlier work of the authors. We then introduce the associated local and global polyhedral fractal zeta functions, and prove that the poles of the global polyhedral fractal zeta function are exactly the same as the Complex Dimensions of the Weierstrass function itself. Contrary to the set of possible Complex Dimensions usually (and classically) obtained by means of tubular neighborhoods, as in an earlier work of ours about the Weierstrass IFD, not only is our result more precise (we obtain the set of actual or exact Complex Dimensions, instead of the set of possible Complex Dimensions), but it also enables us to build a bridge with fractal cohomology, where, for any nonnegative integer m, the mth cohomology group consists of continuous functions which possess a generalized Taylor expansion, with fractional derivatives of orders the underlying—and actual—Complex Dimensions. Then, the aforementioned exact expressions of the polyhedral neighborhoods also enables us to revisit the computation of the box-counting (or Minkowski) dimension of the Weierstrass Curve, in a fully rigorous manner, and thereby to provide a complete and geometric proof of part of Mandelbrot’s conjecture about the fractal dimension of the Weierstrass Curve.
