Subjects studied in number theory

Number theorists study a range of different questions that are loosely inspired by questions related to integers and rational numbers.

Here are some basic topics:

1. Distribution of primes: The archetypal result here is the prime number theorem, stating that the number of primes $\leq x$ is asymptotically $x/\log x$. Another basic result is Dirichlet's theorem on primes in arithmetic progression. More recently, one has the results of Ben Green and Terry Tao on solving linear equations (with $\mathbb Z$-coefficients, say) in primes. Important open problems are Goldbach's conjecture, the twin prime conjecture, and questions about solving non-linear equations in primes (e.g. are there infinitely many primes of the form $n^2 + 1$). The Riemann hypothesis (one of the Clay Institute's Millennium Problems) also fits in here.

2. Diophantine equations: The basic problem here is to solve polynomial equations (e.g. with $\mathbb Z$-coefficients) in integers or rational numbers. One famous problem here is Fermat's Last Theorem (finally solved by Wiles). The theory of elliptic curves over $\mathbb Q$ fits in here. The Birch-Swinnerton-Dyer conjecture (another one of the Clay Institute's Millennium Problems) is a famous open problem about elliptic curves. Mordell's conjecture, proved by Faltings (for which he got the Fields medal) is a famous result. One can also study Diophantine equations mod $p$ (for a prime $p$). The Weil conjectures were a famous problem related to this latter topic, and both Grothendieck and Deligne received Fields medals in part for their work on proving the Weil conjectures.

3. Reciprocity laws: The law of quadratic reciprocity is the beginning result here, but there were many generalizations worked out in the 19th century, culminating in the development of class field theory in the first half of the 20th century. The Langlands program is in part about the development of non-abelian reciprocity laws.

4. Behaviour of arithmetic functions: A typical question here would be to investigate behaviour of functions such as $d(n)$ (the function which counts the number of divisors of a natural number $n$). These functions often behave quite irregularly, but one can study their asymptotic behaviour, or the behaviour on average.

5. Diophantine approximation and transcendence theory: The goal of this area is to establish results about whether certain numbers are irrational or transcendental, and also to investigate how well various irrational numbers can be approximated by rational numbers. (This latter problem is the problem of Diophantine approximation). Some results are Liouville's construction of the first known transcendental number, transcendence results about $e$ and $\pi$, and Roth's theorem on Diophantine approximation (for which he got the Fields medal).

6. The theory of modular (or more generally automorphic) forms: This is an area which grew out of the development of the theory of elliptic functions by Jacobi, but which has always had a strong number-theoretic flavour. The modern theory is highly influenced by ideas of Langlands.

7. The theory of lattices and quadratic forms: The problem of studying quadratic forms goes back at least to the four-squares theorem of Lagrange, and binary quadratic forms were one of the central topics of Gauss's Disquitiones. In its modern form, it ranges from questions such as representing integers by quadratic forms, to studying lattices with good packing properties.

8. Algebraic number theory: This is concerned with studying properties and invariants of algebraic number fields (i.e. finite extensions of $\mathbb Q$) and their rings of integers.

There are more topics than just these; these are the ones that came to mind. Also, these topics are all interrelated in various ways. For example, the prime counting function is an example of one of the arithmetic functions mentioned in (4), and so (1) and (4) are related. As another example, $\zeta$-functions and $L$-functions are basic tools in the study of primes, and also in the study of Diophantine equations, reciprocity laws, and automorphic forms; this gives a common link between (1), (2), (3), and (6). As a third, a basic tool for studying quadratic forms is the associated theta-function; this relates (6) and (7). And reciprocity laws, Diophantine equations, and automorphic forms are all related, not just by their common use of $L$-functions, but by a deep web of conjectures (e.g. the BSD conjecture, and Langlands's conjectures). As yet another example, Diophantine approximation can be an important tool in studying and solving Diophantine equations; thus (2) and (5) are related. Finally, algebraic number theory was essentially invented by Kummer, building on old work of Gauss and Eisenstein, to study reciprocity laws, and also Fermat's Last Theorem. Thus there have always been, and continue to be, very strong relations between topics (2), (3), and (8).

A general rule in number theory, as in all of mathematics, is that it is very difficult to separate important results, techniques, and ideas neatly into distinct areas. For example, $\zeta$- and $L$-functions are analytic functions, but they are basic tools not only in traditional areas of analytic number theory such as (1), but also in areas thought of as being more algebraic, such as (2), (3), and (8). Although some of the areas mentioned above are more closely related to one another than others, they are all linked in various ways (as I have tried to indicate).

[Note: There are Wikipedia entries on many of the topics mentioned above, as well as quite a number of questions and answers on this site. I might add links at some point, but they are not too hard to find in any event.]

Boy, that's a lot of topics for one very simple and interesting (yet complicated) question...

Number Theory encompasses several approaches; at its heart, Number Theory is the study of the properties of the natural numbers, but pretty soon one is led to consider other structures, such as the integers modulo $m$, the rational numbers, complex and real numbers, etc. One could even make a case that the field of complex analysis was born from arithmetic (number-theoretic) considerations.

One might roughly divide number theory into three large swathes: classic or elementary number theory; algebraic number theory; and analytic number theory. (One can also add a fourth: recreational number theory which, to quote Hendrik Lenstra, "is that branch of Number Theory that is too difficult for serious study.")

"Classic" or elementary number theory is the kind of stuff that Fermat was justly famous for: diophantine problems, divisibility questions, the kind of thing in the first section of Gauss's Disquisitiones Arithmeticae, with questions that refer almost exclusively to natural numbers, integers, or rational numbers. Usually, you stick to working with integers, rationals, and integers modulo $n$ for different $n$s. The basic kind of stuff.

Algebraic and analytic number theory were originally distinguished by the kinds of tools that were used in the study of arithmetic problems, but later also by the kinds of questions that were asked (questions that often arose because of the tools one was working on). Analytic number theory uses complex numbers to study arithmetic properties; the original proof of the Prime Number Theorem (a statement about the prime counting function) is a classic example. It uses analytical tools (limits, integrals, real and complex analysis, etc) to study and answers such questions.

Algebraic number theory, on the other hand, uses algebraic (rather than analytical) tools to study problems. A classic example is the study of Fermat's Christmas Theorem (a prime $p$ can be written as a sum of two squares if and only if $p=2$ or $p\equiv 1\pmod{4}$) using the Gaussian integers, $\mathbb{Z}[i]$, or the proofs of quadratic and cubic reciprocity that use the Gaussian and the Eisenstein integers; the study of Gauss sums, etc. A typical object of concern for algebraic number theory is the ring of integers of a finite field extension of $\mathbb{Q}$ (called a number field): start with a finite field extension of $\mathbb{Q}$, $K$, and consider all elements of $K$ that are roots of monic polynomials with integer coefficients. They form a fairly nice ring, with some very nice properties (they are the prototypical example of a Dedekind domain). You also study what one might argue is the algebraic counterpart of the real and complex numbers (relative to the rationals), the $p$-adic numbers $\mathbb{Q}_p$ (you can think of the reals as a way to "complete" the rationals with respect to the absolute value function; you can think of the $p$-adic numbers as a way to "complete" the rationals either with respect to a different kind of absolute value function, or alternatively as a way of making sense of an "infinite sequence of approximations" modulo higher and higher powers of $p$). Algebraic number theory now has its own "higher level offshoot", Arithmetic Geometry, which brings in tools of algebraic geometry to study number theoretic questions (think if "Algebra", "Geometry", "Analysis", "topology, "Number Theory", etc. as 'first-level subjects'; then you have algebraic number theory, algebraic topology, analytic geometry, etc., as 'second-level subjects.' Now we have algebraic arithmetic geometry, a 'third level subject').

In both algebraic and analytic number theory you very quickly are forced to consider irrational numbers, sometimes even transcendental numbers, simply because they are there. The subject of diophantine approximation is an example.

What properties of real numbers are studied in number theory? Well, to be glib, those that are "arithmetically relevant". It's pretty hard to single out certain properties as important and others as not important. Some come up, some do not.

Real numbers are studied from many viewpoints, not merely from number theory. In a sense, the ideas of real numbers (as the real line) are just as old as number theory (if you want to peg Euclid with the basics of number theory, or Diophantus).

As Zev notes, the real numbers (and the complex numbers) have such a rich structure that they are studied from all sorts of viewpoints (topology, order, logic, analysis, algebra, even discrete mathematics). Likewise for the complex numbers.

As to differences in how you study the reals and complex numbers in analysis, in number theory, and in algebra, well, the kinds of questions you are interested in are different. In number theory, it is seldom of any interest that the first order theory of the real numbers with addition and multiplication is decidable (a theorem of Taski's); but that's a pretty big deal in logic. Pretty much like how a study of the Middle Ages might differ depending on whether you are an economist, a historian (a historian interested in Europe; a historian interested in the Middle East; a historian interested in France; etc), a sociologist, an epidemiologist, etc. The different viewpoints all cover the same ground, and will often overlap (transmission of the black death was a key motor of drastic changes in the economies of Europe, giving you a connection between epidemiology and economy), but the viewpoints differ. Given how ubiquitous the real and complex numbers are, you'll encounter them all over the place, playing somewhat different roles in each and provoking somewhat different questions in each.

I am not sure I could confidently claim there is a single area of math to which number theory is not related (perhaps PDEs?) - there are many, many more objects and concepts studied by "number theory" than mere numbers. At any rate: yes, the list of things that come up in number theory includes the real numbers - see this MO question. One of the many reasons given there is that $\mathbb{R}$ is defined to be the completion of $\mathbb{Q}$ with respect to the usual (archimedean) metric, and completions of $\mathbb{Q}$ give us insight into number theoretic questions about prime numbers (note that the other completions of $\mathbb{Q}$ are the $p$-adics $\mathbb{Q}_p$ - there is one for each prime number $p$). So, the real numbers $\mathbb{R}$ are defined in terms of $\mathbb{Q}$, which must be defined in terms of $\mathbb{Z}$; in this sense, none of them can be studied independently from each other. But $\mathbb{R}$ has many structures on it: Hilbert space, measure space, metric space (and hence topological space), field, ordered set, manifold, etc. These structures are emphasized to different extents in different areas of math; but most of these structures on $\mathbb{R}$ are used in number theory in some place or another (note: I don't mean to claim that this is what number theory consists of; only that for each of these structures on $\mathbb{R}$, there exists at least one part of number theory where it is used). Also, $\mathbb{R}^n$ comes up in doing Minkowski's "geometry of numbers"; to give another example, I believe the upper half space of $\mathbb{R}^n$ being acted on by matrix groups is the object of study in areas of number theory I have little knowledge of personally.

In response to your edits: no, what a real analyst studies about real numbers is not a "part of number theory". However, a number theorist (in particular an analytic number theorist) might incorporate some of the results of real analysis about real numbers to prove number theory results. Analytic number theorists also use a lot of complex analysis.

In some ways, $\mathbb{C}$ is an even nicer object than $\mathbb{R}$. It has all of the structures that $\mathbb{R}$ has that I listed above (except for the ordering), but it is algebraically closed which $\mathbb{R}$ is not. This is particularly important to algebra and number theory; it is also evidenced in many ways in complex analysis.

Both of these objects ($\mathbb{R}$ and $\mathbb{C}$) having all of these different structures, all interacting with each other at once, makes them particularly interesting to many different areas of math; they are very singular objects. However, I think the presence of so many structures makes it that much more important to tease apart what exactly a statement about real or complex numbers is using; we may have made a valid statement about $\mathbb{C}$, but perhaps it actually works for any algebraically closed field? Did we need to use the metric of $\mathbb{R}$, or just the topology? And so on.