In Prime Obsession Derbyshire does his best to make the Hypothesis accessible to anyone, however, starting off with almost no assumptions, even explaining powers and logarithms. He builds steadily up from there, however: explaining infinite sequences and series and functions and complex numbers and big oh notation, while introducing the Euler product formula, the Prime Number Theorem, the Zeta function, the Möbius mu function, and "the error term". Along the way he also manages to touch on field theory, algebraic number theory, and links to quantum mechanics and chaotic dynamics.
This mathematics is coupled with an account of the history of the Hypothesis and of associated work in number theory and complex analysis, with mathematical chapters alternating with historical ones. This doesn't restrict itself to Riemann, despite the subtitle, but describes both those whose work led up to his and the most prominent of his successors: notably Euler, Gauss, Dedekind, Hilbert, Landau, Hardy, Littlewood, de la Vallée Poussin, and Hadamard.
Never having delved into analytic number theory, I found a fair bit that was new to me in Prime Obsession. I think it unlikely that many people who start off not knowing what a logarithm is will get to the end of it, but the more accessible material does comes first and the historical account can be followed independently. Anyone with an interest in mathematics who has had some exposure to calculus and complex numbers should enjoy Derbyshire's presentation.