  We use the weighted version of the arithmetic-mean-geometric-mean inequality
to motivate new results about Zaremba's function, $z(n) = \sum_{d|n} \frac{\log
d}{d}$. We investigate record-setting values for $z(n)$ and the related
function $v(n) = \frac{z(n)}{\log \tau(n)}$ where $\tau(n)$ is the number of
divisors of $n$. We show that $v(n)$ takes on a maximum value and we give a
list of all record-setting values for$v(n)$. Closely connected inequalities
motivate the study of numbers which are pseudoperfect in a strong sense.
