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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 re-
cover the shape of a drum from its sound). Here, we consider both types of prob-
lems in the context of drums with fractal (that is, very rough) boundary. We 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 show
that the corresponding inverse spectral problem is intimately connected with the
Riemann Hypothesis, which is arguably the most famous open problem in mathe-
matics and whose solution will likely unlock deep secrets about the prime numbers.
In conclusion, we briefly explain how this work eventually gave rise to a mathemat-
ical theory of complex fractal dimensions (developed by the author and his collabo-
rators), which captures the vibrations that are intrinsic to both fractal geometries
and the prime numbers.
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