  Prime numbers are the building blocks of our arithmetic, however, their
distribution still poses fundamental questions. Bernhard Riemann showed that
the distribution of primes could be given explicitly if one knew the
distribution of the non-trivial zeros of the Riemann $\zeta(s)$ function.
According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator
of which the eigenvalues coincide with the real part of the non-trivial zeros
of $\zeta(s)$. This idea encourages physicists to examine the properties of
such possible operators, and they have found interesting connections between
the distribution of zeros and the distribution of energy eigenvalues of quantum
systems. We apply the Mar{\v{c}}henko approach to construct potentials with
energy eigenvalues equal to the prime numbers and to the zeros of the
$\zeta(s)$ function. We demonstrate the multifractal nature of these potentials
by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope
for further analytical progress.
