A well-known problem in mathematics and physics consists in

understanding how the geometry (or shape) of a musical instrument

affects it sound. This gives rise to two related types of

mathematical problems: direct spectral problems (how the shape of a

drum affects its sound) and inverse spectral problems (how one can

recover the shape of a drum from its sound). Here, we will consider

both types of problems in the context of drums with fractal (that is,

very rough) boundary. We will show, in particular, that one can “hear”

the fractal dimension of the boundary (a certain measure of its

roughness) and in certain cases, a fractal analog of its length. In

the special case of vibrating fractal strings (the one-dimensional

situation), we will show that the corresponding inverse spectral

problem is intimately connected with the Riemann Hypothesis, which is

arguably the most famous open problem in mathematics and whose

solution will likely unlock deep secrets about the prime numbers. In

conclusion, we will briefly explain how this work eventually gave rise

to a mathematical theory of complex fractal dimensions, which captures

the vibrations (or oscillations) that are intrinsic to both fractal geometries and the prime numbers.