# Advice for number theory library

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.

• Obviously this is very subjective, but I consider Selberg's Collected Works as one of the best ways I've spent money. Serre's Works are pretty good too! – Lucia Apr 16 '14 at 15:48
• How much money? – user9072 Apr 16 '14 at 16:36
• Couldn't resist: mathoverflow.net/questions/26267 – Steve Huntsman Apr 16 '14 at 17:16
• Dickson's "History" is indeed great, it basically includes all of number theory that was known at that time, and it is well-organized and readable. It's a fantastic reference for elementary number theory. It is also extremely inexpensive (published by AMS/Chelsea). – Michael Zieve Apr 17 '14 at 3:58
• I also suggest Serre's "A Course in Arithmetic", Ireland-Rosen, Cassels' "Local Fields" (which is great fun), Serre's "Local Fields", Washington's "Cyclotomic Fields", Serre's "Lectures on Mordell-Weil" and "Topics in Galois Theory", Samuel's "Algebraic Theory of Numbers", Lemmermeyer's "Reciprocity Laws", Koblitz's "p-adic numbers, p-adic analysis and zeta functions", Bombieri-Gubler, Cassels-Frohlich, Silverman's various books, etc. – Michael Zieve Apr 17 '14 at 4:06

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

• here you go: Ą Ć Ę Ł Ń Ó Ś Ź Ż ą ć ę ł ń ó ś ź ż -- cut and paste to your heart desire. Say: Władysław. And indeed, Władysław Narkiewicz has published an impressive, high quality array of books in number theory (outside the history of number theory), including one on algebraic number theory. – Włodzimierz Holsztyński Apr 17 '14 at 18:26
• BTW, it's "y", not "i", in Władysław. – Włodzimierz Holsztyński Apr 17 '14 at 18:27
• Thank you. I will fix the i/y problem. You are welcome to modify the rest and remove the parenthetical comment. (I've never studied the relevant languages, and prefer poor Romanization by myself or correction by someone else more knowledgeable.) Also, copy-paste has given me problems lately. Gerhard "May Switch Back To Pencils" Paseman, 2014.04.17 – Gerhard Paseman Apr 17 '14 at 18:39
• Gerhard, thank you for kind response. In addition to the Polish Władysław there is also Vladislav (with "i" this time) in several Slavic languages. You simply, if temporarily, have created a panslavic name Wladislaw. A related similar name, which has somewhat different meaning though, is Russian Vladimir or its Polish equivalent Włodzimierz. – Włodzimierz Holsztyński Apr 17 '14 at 18:53

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

Gauss, Hasse,

and their correspondence

Hasse-Artin, Serre-Tate (worth the wait)

Finally, the history

Weil, Dieudonné

• This is too unspecific to be actually helpful in my opinion. – user9072 Apr 17 '14 at 15:50
• @Chandan Singh Dalawat: could you edit your post more horizontally? – Włodzimierz Holsztyński Apr 17 '14 at 16:09
• @Chandan Singh Dalawat: what would be a book by Hermann Minkowski? Do you have one in your library? I know only about the 2-volume monograph by Harris Hancock: Development of the Minkowski Geometry of Numbers. – Włodzimierz Holsztyński Apr 17 '14 at 16:49
• @WlodzimierzHolsztynski re Minkowski's books the most pertinent to the question at hand should be 'Diophantische Approximation' and 'Geometrie der Zahlen' (Geometry of Numbers). But he wrote other books too, especially on theory of relativity. – user9072 Apr 17 '14 at 20:46
• @quid--thank you. So, it was obviously in the preEnglish era; and not all books were translated by Soviets (or some were out of print). – Włodzimierz Holsztyński Apr 17 '14 at 21:44

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.