  We study de Haas-van Alphen oscillations in a marginal Fermi liquid resulting
from a three-dimensional metal tuned to a quantum-critical point (QCP). We show
that the conventional approach based on extensions of the Lifshitz-Kosevich
formula for the oscillation amplitudes becomes inapplicable when the
correlation length exceeds the cyclotron radius. This breakdown is due to (i)
non-analytic finite-temperature contributions to the fermion self-energy (ii)
an enhancement of the oscillatory part of the self-energy by quantum
fluctuations, and (iii) non-trivial dynamical scaling laws associated with the
quantum critical point. We properly incorporate these effects within the
Luttinger-Ward-Eliashberg framework for the thermodynamic potential by treating
the fermionic and bosonic contributions on equal footing. As a result, we
obtain the modified expressions for the oscillations of entropy and
magnetization that remain valid in the non-Fermi liquid regime.
