Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I decided to use the money to build a small library in my office. I'm a number theorist. Specifically I'm in analytic number theory. But I like all kinds of number theory and want to learn more of the subject, so I want a fairly well rounded library. The catch is that when it comes to reading, I am attracted to explicit / classical language, rather than abstract / very modern. Please suggest a list of good books for my library.
I have not seen mention of Władysław Narkiewicz (feel free to correct the accents and other marks) and his books on the history of number theory, as well as Ribenboim's texts on certain Diophantine equations. This could be for the lending portion of your library, for those who want to get their toe in the door without having much more than mathematical maturity and a stomach for notation. I have not checked the bibliographies, but I imagine they would help as much as an index you made yourself for your library.
Gerhard "Not A Number Theorist, Yet" Paseman, 2014.04.17
Books on Number Theory :
Legendre, Gauss, Jacobi, Dirichelet-Dedekind, Kronecker, Minkowski, Hilbert , Hensel, Hecke, Weyl, Hasse, Artin, Artin-Tate, Lang, Weil, Iwasawa, Serre, Serre, Serre, ..., Tate, Cassels, Swinnerton-Dyer, Manin-Panchiskin, Neukirch, Borevich-Shafarevich, Fröhlich-Taylor, Kato-Kurokawa-Saito, Goldfeld...
If you still have some money, go in for the collected papers of arithmeticians
Gauss, Dirichelet, Eisenstein, Kummer, Riemann, Kronecker, Minkowski, Hilbert, Ramanujan, Takagi, Hasse, Hecke, Siegel, Artin, Weil, Iwasawa, Shafarevich, Serre, Manin, Tate (whenever they're published)
and conference proceedings, among them
Cassels-Fröhlich, Fröhlich, Cornell-Silverman, Borel-Casselmann, Szpiro, Jannsen-Kleiman-Serre, Cornell-Silverman-Stevens, Conrad-Rubin, Popescu-Rubin-Siverberg, Sarnak-Shahidi, Darmon-Ellwood-Hassett-Tschinkel, 2009 Clay Summer School
and arithmeticians' diaries
and their correspondence
Hasse-Artin, Serre-Tate (worth the wait)
Finally, the history
I don't know the newer books, but otherwise a classic within the geometry of numbers is the "geometry of numbers" by C.G.Lekkerkerker (as presented on a finite width title page).
Years before that there was an earlier classic "An Introduction to the Geometry of Numbers" by J.W.S.Cassels. (I used to have its Soviet translation, in Russian of course).
There are already several wonderful monographs mentioned above, and I may mention a couple more in another answer too (below this one). However, if there was just one to choose, I would select one here without any hesitation, be it on number theory or from all mathematical books.
This book is not too well spread around because there is only a limited number of copies of it. (Also, it has a strange peculiarity which I will mention in a moment). The special (for me) book is
Emil Artin, Theory of Algebraic Numbers
The book is based on Artin's notes taken by Gerhard Wurges, 1956/7. It was translated into English and distributed by George Striker. Both deserve gratitude for their contribution.
The impressive thing (one of them) is that the text is completely void of any public relation chat. Any introductory remarks to the consecutive segments of the text are minimal, almost non-existent. And still, the text rolls smoothly and naturally. Of course all experts can appreciate the beautiful construction, discretely axiomatic, with axioms taken almost for granted.
Now about peculiarity. You may see that at one moment a statement is called a theorem where this is really not any theorem to be called by such a proud name. And then soon you can see an unannounced formulation which actually is a theorem. Well, at least at one time the note taker. as well as the translator, didn't concentrate too well. Despite a somewhat imperfect medium, the monograph is beautiful all the same.
There was just one not strictly mathematical omission in the book, of not crediting at all Stanisław Mazur for his theorem about the division Banach algebras. But then, Mazur was hardly ever fully credited for his wonderful result anyway.