A length n cosine sum is an expression of the form cos a1✓+· · ·+cos an✓ where a1 < · · · < an are positive integers, and a length n Newman polynomial is an expression of the form za1 + · · · + zan where a1 < · · · < an are nonnegative integers. We define (n) to be the largest minimum of a length n cosine sum as {a1, …, an} ranges over all sets of n positive integers, and we define µ(n) to be the largest minimum modulus on the unit circle of a length n Newman polynomial as {a1, …, an} ranges over all sets of n nonnegative integers. Since there are infinitely many possibilities for the aj, it is not obvious how to compute (n) or µ(n) for a given n in finitely many steps. Campbell et al. found the value of µ(3) in 1983, and Goddard found the value of µ(4) in 1992. In this paper, we find the values of (2) and (3) and nontrivial bounds on µ(5). We also include further remarks on the seemingly di cult general task of reducing the computation of (n) or µ(n) to a finite problem.
