- ★★★★★ The Music of the Primes by Marcus du Sautoy: Probably the best account on mathematics written for a broad audience. Captivating and insightful. If you're interested in the topic, there's basically no way around reading this. And for those that are afraid of maths: it's virtually free of formulae, but explains in great metaphors the beauty of the prime numbers and the significance the Riemann Hypothesis plays for them.
- ★★★★★ Prime Obsession by John Derbyshire: More mathematically ambitious than The Music of the Primes, but still targeted at the general public (mostly successfully, I think), this is the natural follow-up if Marcus du Sautoy got you hooked up, but left you alone with the maths. John Derbyshire does not flinch from doing actual maths (though he keeps calculus to a minimum), and generally succeeds in explaining the mathematical background of the Riemann Hypothesis to non-mathematicians -- this alone makes the book worth reading. The anecdotes, stories, and historical notes on the Riemann Hypothesis are a nice bonus, including to the most comprehensive account on Bernhard Riemann's life known to me.
- ★★★★☆ Riemann's Zeta Function by H.M. Edwards
- ★★★☆☆ Dr. Riemann's Zeros by Karl Sabbagh
Everything You Always Wanted to Know About Primes (But Were Afraid to Ask)