# How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest in, and now I'm currently exploring both fields as much as possible on my own and it is number theory that I'm finding hard to explore( on my own).

I have done everything , I came across including moderate level of problem solving related to elementary number theory and as I wish to study it further at a more advanced level, I searched on net, asked to people for advice on how can I proceed to do that, but till now have come to know only the basic stuff like: At advanced level it is divided into 2 parts: algebraic and analytic and to study analytic you should have knowledge of complex analysis and basic abstract algebra about algebraic NT.

At this point , I do not have much knowledge in abstract Algebra and know even less in complex analysis, so here goes my first question: Am I right in first clearing my concepts in basic complex analysis ( for analytic) and algebra( for algebraic). If yes, then exactly what or how much should I know in both of these subjects to study these two branches, at advanced level. and, which books I can follow for them that are not too hard for a beginner for me?

Next, I looked into the classic text of analytic Number theory by Tom Apostol, often recommended to study Analytic number theory and found the first few chapters appropriate for a beginner, requiring no pre requisites of analysis. So, my next question is: at what point while I'm doing this or any other book in ANT would I require knowledge in analysis( complex).

Last question: The method I'm following in studying and exploring new fields of interest in Mathematics ( i.e., of studying in detail both these fields at the same time, until I reach a point when I have found out which of these two interest me most, and study that one further). Is it right? If not, then how should I study instead?

• It is always good, imo, to explore new stuff in any realm of human knowledge as long as one keeps in mind not to fall in desperation for not understanding all as one must study more. You'll probably need analysis (both real and complex) pretty soon, relatively, in number theory, but you can go on all the time as long as you feel comfortable with what you read. You'll probably need to wait until 3rd-4th undergraduate year to have the tools to understand most of Apostol's book, but you may keep on peeking at it as much as you want. – DonAntonio Mar 1 '14 at 11:10
• "Right now, I can't think of more questions that I had to ask, so I'll just stop here." It is the norm of Math.SE to ask one Question that can be answered in a concise, specific fashion. Thus, asking what parts of Apostol's book require an undergraduate course in complex analysis is reasonable. Asking broadly about "exploring new fields of interest in Mathematics" may be genuinely interesting but not appropriate for this site. – hardmath Mar 1 '14 at 11:48
• I think, Ramanujan's works on Number Theory will be of great interest for you. – gaurav Mar 1 '14 at 11:53

If you have already learnt group theory, I may suggest you to go through the book 'theory of algebraic numbers' by Pollard & Diamond. It's a really good treatise to start off. You don't even need to know the definition of ring to read this book. Everything is given there in a very well setup. After having finished that book, you may pay a look at 'A Classical Introduction to Modern Number Theory' by Ireland & Rosen.

Some few remarks to your questions:

Am I right in first clearing my concepts in basic complex analysis (for analytic) and algebra( for algebraic)?

Yes, you're right. A solid basis of the fundamental concepts of algebra and complex analysis at least at a moderate level will enrich your studies in number theory.

Let's have a look for example at Algebraic Number Theory and Fermat's last theorem a nice and accessible starter to delve into algebraic number theory. The authors Ian Stewart and David Tall wrote the book explicitly with the intention to be self-contained with all the necessary algebra developed within the book. Consequently it starts with the first section I.1 Algebraic Background: Rings and Fields, Field extension, Free Abelian Groups, etc.

I appreciate this book and recommend it as proper starting point for beginners. But despite being self-contained with respect to algebra you will nevertheless gain considerable payoff when you already have a certain familiarity with abstract algebra. You will make a better progress because you can put the focus to the more number theoretic aspects. When learning abstract algebra only en passant while reading algebraic number theory you will hardly get a proper impression of the used algebraic framework reducing so aha experiences and reducing pleasure!

A nice introductory text in abstract algebra is A First Course in Abstract Algebra from John B. Fraileigh.

The same arguments hold for complex analysis with respect to analytic number theory. You already had a look into Apostol's Introduction to Analytic Number Theory. His second volume Modular Functions and Dirichlet Series in Number Theory dealing with elliptic functions, modular forms, etc is complex Analysis right from the beginning.

One of my favorites in complex analysis is Walter Rudins Real and Complex Analysis. You may first look into his classic The Principals of Mathematical Analysis to get a better impression about his writings.

At what point while I'm doing this or any other book in ANT would I require knowledge in analysis( complex)?

In my opinion, whenever you study analytical properties of functions resp. series a sound knowledge of basics in complex analysis is helpful. This is independent from studying number theory. A very simple example: If you look at the series expansion of $$\frac{1}{1+x^2}$$ the radius of convergence is equal to $$1$$. You can see it when considering the natural domain of the function.

Last question: The method I'm following in studying and exploring new fields of interest in Mathematics ( i.e., of studying in detail both these fields at the same time, until I reach a point when I have found out which of these two interest me most, and study that one further). Is it right? If not, then how should I study instead?

Your approach looks good and promising. Just to make some things easier for you, you could visit parallel to your number theoretical studies at least a one semester course (better two) in algebra and/or complex analysis combined with the tutorials (lots of exercises are crucial).

• +101 Excellent and thorough answer. But I think big Rudin may be a bit much for someone without previous experience in complex analysis. Maybe a text along the lines of Bak & Newman's Complex Analysis would be a better first text for someone interested in number theory. – Gamma Function Jul 24 '14 at 20:25
• @GammaFunction: Thank you for the bounty and - yes big Rudin is tough material for a beginner. At first I wanted instead recommend Conways classic Functions of One Complex Variable then Remmerts Theory of Complex Functions since I'm also fond of reading about historical connections. – Markus Scheuer Jul 24 '14 at 21:23
• @GammaFunction: (cont.) But then I thought since Fraileighs book teaches Algreba so extraordinarily gentle that a challenging contrast like Rudins book is a good stimulation for a smart student. In fact I tried to mitigate the challenge with my hint to Rudins Principles. By the way my tough challenge many years ago was the first volume of Dieudonnes Foundations of Modern Analysis :-) – Markus Scheuer Jul 24 '14 at 21:24
• @MarkusScheuer: Thanks a lot for your great answer! I 'll certainly keep in mind your advice to my present and future studies in the area. – Mojojojo Aug 1 '14 at 16:04
• @shrey: You're welcome! Best regards, – Markus Scheuer Aug 1 '14 at 17:16