  We study the thermodynamics of the maximally supersymmetric Yang-Mills theory
with gauge group U(N) on R x S^3, dual to type IIB superstring theory on AdS_5
x S^5. While both theories are well-known to exhibit Hagedorn behavior at
infinite N, we find evidence that this is replaced by Lee-Yang behavior at
large but finite N: the zeros of the partition function condense into two arcs
in the complex temperature plane that pinch the real axis at the temperature of
the confinement-deconfinement transition. Concretely, we demonstrate this for
the free theory via exact calculations of the (unrefined and refined) partition
functions at N<=7 for the su(2) sector containing two complex scalars, as well
as at N<=5 for the su(2|3) sector containing 3 complex scalars and 2 fermions.
In order to obtain these explicit results, we use a Molien-Weyl formula for
arbitrary field content, utilizing the equivalence of the partition function
with what is known to mathematicians as the Poincare series of trace algebras
of generic matrices. Via this Molien-Weyl formula, we also generate exact
results for larger sectors.
