  We investigate the localization of waves in aperiodic structures that
manifest the characteristic multiscale complexity of certain arithmetic
functions with a central role in number theory. In particular, we study the
eigenspectra and wave localization properties of tight-binding Schr\"{o}dinger
equation models with on-site potentials distributed according to the Liouville
function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and the Legendre
sequence of quadratic residues modulo a prime (QRs). We employ Multifractal
Detrended Fluctuation Analysis (MDFA) and establish the multifractal scaling
properties of the energy spectra in these systems. Moreover, by systematically
analyzing the spatial eigenmodes and their level spacing distributions, we show
the absence of level repulsion with broadband localization across the entire
energy spectra. Our study introduces deterministic aperiodic systems whose
eigenmodes are all strongly localized in realistic finite one-dimensional
systems and provides opportunities for novel quantum and classical devices of
particular importance to cold-atom experiments in engineered speckle potentials
and enhanced light-matter interactions.
