This is my favorite of all time number theory books. It does a lot of pretty high-powered number theory starting with things that a well-prepared and committed math undergrad should know: strong algebra and decent background in real analysis: no previous number theory required.

Prof. Rademacher starts naively with Farey fractions (fractions with bounded denominator arranged sided polygon. Gauss' insights are displayed clearly, and explanations and motivation are supplied everywhere. Rademacher is not afraid to display computations at length, but these are organized in such a way as to make the general principles crystal clear. This section of the book ends with a nice proof of Quadratic Reciprocity, using "finite Fourier series".

This is followed contained treatment of Dirichlet's theorem on primes in arithmetic progressions, including a quick intro to group characters and L-series. This is a great introduction to the methods of analytic number theory.

Finally, the book concludes with a hard-to-find-elsewhere account of V. Brun's theorem on prime pairs: the sum, of the reciprocals of primes that differ by 2, converges. This is one of the hardest chapters in the book, but the result and the insight gained in proving it is well worth the effort.