Textbooks for studying Riemann hypothesis

Most books on analytic number theory mention the Riemann zeta function in one way or another. In fact, there is a nice Dover book (so it's fairly cheap) called Riemann's Zeta Function by Edwards: https://www.amazon.com/Riemanns-Zeta-Function-Harold-Edwards/dp/0486417409 -- that's a place to start anyway. It goes through the basics, and beyond, and only presupposes some knowledge of one variable complex analysis.

I recognize that this question is old enough that the OP has presumably already found the book he was looking for, but in case it is helpful to others I just wanted to note that Barry Mazur and William Stein have recently published an excellent book (Prime Numbers and the Riemann Hypothesis) which aims to explain the statement and significance of the Riemann Hypothesis to readers with very minimal backgrounds. The majority of the book assumes only a basic knowledge of calculus while the end assumes a bit of complex analysis.

This past January I mentored a reading course in which a student that had just finished taking Calc 2 worked through the book. (I mentioned the Riemann hypothesis briefly during our discussion of p-series and worked out its Euler product to illustrate the connection to primes.) There were a few points that I had to explain to him, but overall my impression was that he enjoyed the book quite a bit and that it really helped him appreciate the importance of the problem. I should also point out that the book is quite inexpensive (~$20).