  We give new upper and lower bounds on the minimax sample complexity of
differentially private mean estimation of distributions with bounded $k$-th
moments. Roughly speaking, in the univariate case, we show that $n =
\Theta\left(\frac{1}{\alpha^2} +
\frac{1}{\alpha^{\frac{k}{k-1}}\varepsilon}\right)$ samples are necessary and
sufficient to estimate the mean to $\alpha$-accuracy under
$\varepsilon$-differential privacy, or any of its common relaxations. This
result demonstrates a qualitatively different behavior compared to estimation
absent privacy constraints, for which the sample complexity is identical for
all $k \geq 2$. We also give algorithms for the multivariate setting whose
sample complexity is a factor of $O(d)$ larger than the univariate case.
