  A length $n$ cosine sum is an expression of the form $\cos a_1\theta + \cdots
+ \cos a_n\theta$ where $a_1 < \cdots < a_n$ are positive integers, and a
length $n$ Newman polynomial is an expression of the form $z^{a_1} + \cdots +
z^{a_n}$ where $a_1 < \cdots < a_n$ are nonnegative integers. We define
$-\lambda(n)$ to be the largest minimum of a length $n$ cosine sum as
$\{a_1,\ldots,a_n\}$ ranges over all sets of $n$ positive integers, and we
define $\mu(n)$ to be the largest minimum modulus on the unit circle of a
length $n$ Newman polynomial as $\{a_1,\ldots,a_n\}$ ranges over all sets of
$n$ nonnegative integers. Since there are infinitely many possibilities for the
$a_j$, it is not obvious how to compute $\lambda(n)$ or $\mu(n)$ for a given
$n$ in finitely many steps. Campbell et al. found the value of $\mu(3)$ in
1983, and Goddard found the value of $\mu(4)$ in 1992. In this paper, we find
the values of $\lambda(2)$ and $\lambda(3)$ and nontrivial bounds on $\mu(5)$.
We also include further remarks on the seemingly difficult general task of
reducing the computation of $\lambda(n)$ or $\mu(n)$ to a finite problem.
