Motivation behind Analytic Number Theory

Question

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now started reading Davenport's 'Multiplicative Number Theory'. Without going into too much detail, I found that, while the results used some common techniques in their proofs, they were otherwise quite independent. Since I am relatively inexperienced, I found that quite strange, considering that most subjects I have read so far (real and complex analysis, abstract algebra, measure theory, functional analysis, algebraic topology) seem to have a coherent development, rather than simply be a collection of problems that have been solved using roughly similar machinery.

I was wondering if there is an overarching idea behind the study of analytic number theory (classical, as well as sieve methods), any specific open problems that motivated past research in the field, and if there is any textbook that treats it from that perspective, rather than just a collection of interesting problems. For instance, although I know very little about it, my professor once told me about the Langlands Program and said that the current goal of several mathematicians working in areas of algebraic number theory and automorphic forms to resolve the conjectures of that program.

Also, unlike many other areas, I couldn't effectively use the approach many of my professors seem to recommend, of reading the theorem and attempting to prove it by myself. I am inclined to believe that it is my own shortcomings that prevent this approach, but if it is a part of a wider trend, and I am reading it "wrong", for want of a better phrase, I would like to know the same, and would like to know how exactly such a subject is to be most efficiently learnt.

I haven't quite managed to phrase the question as well as I had hoped to, so, if you have any replies to the title in itself, any motivation towards a coherent understanding of the subject, then, I would be much obliged to you for your input/advice.

I wasn't quite sure what tag to use, and so, chose what I thought was most appropriate. I hope this will not be an issue.

Answer 1

I will go out on a limb and say that in my opinion, it is the norm, rather than the exception, for a branch of mathematics to be a collection of results that we can prove using the techniques we know, rather than a beautiful coherent theory. As a student, one is exposed to a biased sample—the areas of mathematics that have been sufficiently developed to form a systematic theory are precisely the areas that lend themselves to the "coherent development" that you mentioned. This can create an illusion that all areas of mathematics are like that.

If humanity were collectively much more intelligent than it is, our textbook on analytic number theory would probably first prove "Theorem 1: The Generalized Riemann Hypothesis" and then go on to prove all the results that are predicted by pretending that the prime numbers are random but that don't immediately follow from GRH. This would be a beautifully coherent treatment of the subject. But since we're so woefully far away from being able to prove such things, such a textbook is simply not an option.

Having said that, I do agree with you that some textbooks could present more forest and fewer trees than they do. One text I'd recommend is Donald Newman's Analytic Number Theory. Newman rightly emphasizes that the first thing you have to get thoroughly comfortable with is the idea that you can study numbers by studying their generating functions. Newman goes through several examples, starting with easy ones and working up to harder ones. Getting completely comfortable with generating functions is so important that I'd even recommend that you spend some time with books such as Wilf's generatingfunctionology or Flajolet and Sedgewick's Analytic Combinatorics just to get a feeling for what generating functions can do for you. Wilf and Flajolet–Sedgewick are interested in combinatorics rather than number theory and so focus on ordinary generating functions rather than Dirichlet series, but any increase in your comfort level with generating functions will pay dividends in your study of analytic number theory. You will probably recall that Apostol's book spends a lot of time studying different generating functions and formulating many different equivalent statements of (for example) the prime number theorem. I found all these manipulations mysterious and unmotivated the first time I encountered them, but once you understand the value of packaging information in a generating function in different ways, the manipulations will seem less baffling.

The next big idea in analytic number theory is that complex analysis, in particular the study of zeros and singularities, gives you asymptotic information about generating functions. Again, the Flajolet–Sedgewick book gives many examples of this general principle for ordinary (and exponential) generating functions, where it's easier to get a feel for why there is a connection between singularities in the complex plane and asymptotics. For analytic number theory one must also study Dirichlet series, where there are more technical difficulties, but at a high level, the basic idea is the same. I like the treatment in Serre's Course in Arithmetic, which proves in a unified way some basic properties that hold for both Dirichlet series and ordinary power series, and makes it clear where the analogy between the two starts to break down. (Also Serre's treatment of Dirichlet's theorem on arithmetic progressions is very clean; historically, Dirichlet's theorem was arguably the first big result of analytic number theory, so it's a good theorem to master thoroughly at an early stage of your study.) The treatment of Dirichlet series in Montgomery and Vaughan's Multiplicative Number Theory is also excellent.

At some point I think it is worth studying Riemann's paper on the zeta function. The book by Harold Edwards Riemann's Zeta Function provides an excellent guided tour of Riemann's paper. John Derbyshire's book Prime Obsession is also worth looking it; it's semi-popular but gives enough technical details for you to understand what Riemann's exact formula actually says. I think that a thorough understanding of Riemann's exact formula will be a big step forward in getting the bird's-eye view of analytic number theory that you're seeking.

Finally, for a short overview of the whole subject, I think that Andrew Granville's article in the Princeton Companion to Mathematics is hard to beat. It goes into subjects such as sieve theory and prime gaps that I haven't mentioned here.

Answer 2

Wladyslaw Narkiewicz has published more than one book on the historical development of number theory. I recommend skimming them once now, so that you get some idea of a partial map of the analytic number theory terrain, and then go back to them after some time studying the subject. I am not a professional student of analytic number theory, but my exposure to various topics in number theory gives me great appreciation for these extensive surveys. If you know you are going to spend time with ANT, spend a few hours now perusing his books The Development of Prime Number Theory and Rational Number Theory in the 20th Century (and other books of his that catch your interest). This will be time well spent.

Gerhard "Has Them In His Backpack" Paseman, 2019.02.14.