The link between prime numbers and the Riemann hypothesis lies in analytic number theory  indeed the Prime Number Theorem was first proved using complex analysis.
The only book I've read on this is T. Apostol, "Introduction to Analytic Number Theory" which contains the standard dose of elementary number theory, proves the PNT and discusses the Riemann Hypothesis (but not in a great deal of depth). The calculus course will help, some prior knowledge of complex analysis is helpful too. The book is very readable, insofar as such books ever are.
Dave
