How to learn to like number theory

[nonequilibrium]

Hello,

I'm currently taking a course in number theory, and I usually enjoy every branch of pure mathematics, but somehow number theory is not really exciting me. It's hard to pin-point why exactly... Perhaps the following two feelings:
- It's hard to see a real structure when trying to tackle a given problem (i.e. a certain exercise). It seems to be rather dependent of accidentally thinking of a certain trick.
- The theorems are often awfully concrete (talking specifically about certain numbers), not giving off an aura of blissful generality through abstraction (as opposed to any other field of pure mathematics, really).

Am I perceiving things the wrong way? Or is it correct and do I simply not appreciate these characteristics enough whilst others do? Or does it simply take some time to get into the right atmosphere for these things? (For example I remember first struggling with Abstract Algebra, I wasn't connecting with it in the beginning either, but now I do; maybe the same thing must happen with Number Theory?)

What is Number Theory to you? And was it love on first sight? What is your stance towards is?

PS: I'm in first place a physicist, so if anyone knows of relevance for number theory for physics, do share, but I'm pretty confident that there isn't (not that relevance for physics is necessary for me to enjoy math!)

(Note: I realize that it is very useful for cryptography, but those things don't interest me and probably never will, so I can't really turn to that for motivation.)

EDIT: It might matter, or it might not, but my "favorite" area of pure math is analysis. Then either algebra or topology. And then geometry. I might like the foundations of math (I like foundations, generally), but I haven't really been exposed to it formally yet. At the bottom I would currently place number theory, but hopefully this will change somewhat...

 

[dodo]

Hi, mr. vodka,
I'm only a student and not on physics, but here is my opinion for what is worth.

Curiosly enough, A) I have exactly the same impression and B) that is *precisely* what I love of the subject! Many seemingly simple issues appear to lack a larger, underlying scheme; different issues appear unconnected. I can't imagine a more luring reason to explore them. I am continually thinking, "someone should put some order in this mess".

Further topics you may study later will have connections to other math branches, particularly to group theory.

 

[DonAntonio]

Perhaps it is your instructor who's making Number Theory, also known as "the queen of mathematics", not interesting... a shame, since it is a fascinating, astonishingly beautiful subject. By the way, huge chunks of number theory use analysis in a very strong way, both real and, more importantly many times, comple analysis, so if you continue on this you shall be meeting lots of analysis soon (e.g., in Dirichlet's theorem on arithmetic sequences, Riemann's Zeta Function, The prime Number theorem and a large etc.)

 

[20Tauri]

My favorite professor says that number theory is "the very best math there is--don't let anyone tell you otherwise," and I am inclined to agree with her. I liked learning to do proofs with something a little more concrete than, say, vector spaces, and the cryptography unit was really fun, and I flipped out over how cool cyclic groups in prime moduli were. I really enjoyed all of it--number theory is my favorite math class so far.

I have been told that since I liked number theory, I'll probably like abstract algebra. Maybe you'll have sort of the reverse? Since you came around to liking algebra, you'll find something you enjoy in number theory?

 

[chiro]

One way you can think of number theory with congruences is to think of it as a integer form of periodic analysis.

The idea of this discrete periodic analysis allows us to think about quantization issues when we need to analyze periodic systems and since periodic systems exist everywhere in nature (I would go far as to say that nature is 'based' on this principle in many ways), then by understanding the properties of general periodic discrete systems, that one can also understand in some real context the issues that come about when you have to choose a quantization model for something that has known periodic processes.

It's not just only about primes, it's also about the analysis of periodicity in a truly quantized context.

 

[mathwonk]

have you looked at the zeta function, and its connection with the euler product? and more generally at dirichlet series? analytic number theory is one of the deepest parts of the subject.

 

[morphism]

To echo mathwonk and DonAntonio, stop and think about this for a second: why in the world does analysis (let alone complex analysis) tell you anything about number theory? The answer is of course not a priori obvious, and in fact remains quite mysterious to this day. If you approach analytic number theory with the motivation of answering this question then you might actually find yourself enjoying it.

For more concrete advice, I highly recommend that you try to learn Dirichlet's proof of his theorem on primes in arithmetic progressions (don't get bogged down by the error term or density results: just learn about the existence portion of the theorem). Then try to mimic his proof to show that there are infinitely many primes of the form 4k+1 and 4k+3. You will most likely be extremely impressed with his techniques; I know I was when I first carried this out.

If you're too lazy to go through Dirichlet's entire proof, then I highly recommend the first section of Chapter VII of Knapp's Elliptic Curves. He very lucidly describes the proof in the case of 4k±1.

Of course, Dirichlet's proof of his theorem was motivated by (one of?) Euler's proofs of the infinitude of primes, the one where he factors the zeta function into (what we now call) an Euler product. Dirichlet noticed that the same kind of argument is possible if we replace the numerator in the zeta function (the number 1) with something that's multiplicative, i.e. with a certain homomorphism, now called a Dirichlet character.

You can think of a Dirichlet character mod n as a multiplicative homomorphism from the multiplicative group (Z/nZ)* into the nonzero complexes C*. But of course C*=GL_1(C), so you can think of a Dirichlet char mod n as a one-dimensional representation of the group (Z/nZ)*. And what is this group if not the Galois group of the nth cyclotomic extension Q(zeta_n)/Q? So by studying representations of this Galois group we seem to be obtaining arithmetic information. Dirichlet characters are thus an en entry point to the large industry of Galois representations, the ideas of which were instrumental in, e.g., Wiles's proof of Fermat's Last Theorem.

Since I mentioned the cyclotomic extension Q(zeta_n)/Q, let me mention one really nice thread in this vein: it's a celebrated theorem of Kronecker and Weber that every abelian extension K of the rationals Q (i.e. every Galois extension K/Q with abelian Galois group) is contained in some Q(zeta_n). One can think of zeta_n as exp(2pi i/n), so the Kronecker-Weber theorem may be viewed as saying that we obtain every abelian extension of Q as sitting inside Q(exp(2pi i z)) for an appropriate evaluation of z (namely z=1/n). The function exp(2pi i z) is of course a very nice "transcendental" function, and Kronecker wondered if we could obtain all "nice" extensions (not of Q -- but of more general fields) by similarly adjoining to the base field certain values of similar transcendental functions. He called this his Jugendtraum -- the dream of his youth. Try to read up on this -- it's really fascinating. It is essentially Hilbert's 12th problem, and will lead you to a lot of interesting number theory, most notably class field theory, which many number theorists consider one of the crowning achievements of 20th century math.

Of course once you start reading about class field theory, you will invariably be led back to the world of Galois representations, and then perhaps to the Langlands program and the theory of Shimura varieties, which are two extremely active areas of contemporary research.

Anyway, the moral is: number theory is a very large subject, full of deep and beautiful ideas. I'm sure anyone who has any interest in math can easily find some little corner of the subject that they would enjoy; they just have to look in the right spots...

 

[mathwonk]

here is an excerpt from my review of Rieman's works:

Prime numbers
on the number of primes less than a given magnitude.
i have gained the following impression from my reading and would appreciate corrections from experts.

it seems that first euler noticed that, assuming the factorization of natural numbers into primes, that speaking purely formally, one has the sum of all the natural numbers n equal to the product of the infinitely many factors
(1+2 + 2^2 + 2^3 +...)(1+3+3^2+3^3+...)(...)(1+p+p^2+p^3+...)(... ..).

To try to render this meaningful, one can instead consider reciprocals, since now

(1+1/2+1/2^2+...) = 1/(1 - [1/2]),..., (1+1/p+1/p^2+...) = 1/[1 - 1/p], ...
so now at least all the factors are finite, and we may hope that the product of the factors
1/[1 - 1/p] for all primes p, equals the sum of the reciprocals of the natural numbers

1 + 1/2 + 1/3 + 1/4 +... however still both these infinite expressions diverge.then euler thought to use exponents. indeed he knew the sum of the squares of the reciprocals of natural numbers equalled pi^2/6, etc...,

so he looked at the equation PRODUCT (over all primes p) of 1/[1 - (1/p)^s]
= SUM over all natural numbers n, of 1/n^s. he considered it principally for integers s.
then perhaps legendre looked at this expression as a function of a real variable s, and noted it converges for any s > 1.

then riemann took it up and naturally for him asked for its fullest range of definition. thus he considered complex values of s, obtaining a complex meromorphic function called the zeta function.

i.e. for riemann a complex holomorphic function is determined globally by its values in any region, and moreover, his overriding point of view was that a meromorphic complex function is best understood by its zeroes and poles.

hence here is a function which is wholly determined by an expression involving only prime numbers, so its behavior reflects the nature, i.e. location, of the prime numbers, but which is best understood by considering its zeroes and poles (of which it has none for Re(s) > 1). thus the curious fact that s = 1 is a point where the original formula makes no sense, is replaced by the intersting fact that s =1 is the unique pole of the global extension of this function. hence the other crucial points must be the zeroes of the function, none of which are visible until Riemann's extension of the function.

now the problem riemann considered was that of determining roughly the number of prime numbers that are less than a given number x = pi(x).

according to edwards, euler's observations on the rate of divergence of the series 1+ 1/2 + 1/3 + 1/5 +...+1/p+... can be phrased as saying the density of primes is 1/log(x).

Then Gauss conjectured that the number pi(x) was well approximated by Li(x) = the integral from 0 to x of dt/ln(t), (up to a small constant).

now riemann's paper attempted to improve this approximation for pi(x) as follows:

Using his zeta function, and various manipulations of integral expressions for it, Riemann claimed to show that in fact Gauss's estimate Li(x) approximates more closely the number

Li(x) (roughly) = pi(x) + (1/2)pi(x^[1/2]) + (1/3)pi(x^[1/3]) + ...

i.e. not just the number of primes less than x but also 1/2 the number of squares of primes, plus 1/3 the number of cubes of primes, ...

hence he says that Li(x) overestimates pi(x) by a term of order of magnitude x^(1/2).

By solving the approximate equation above for pi(x) he obtains a formula approximating pi(x) (roughly) =
Li(x) - (1/2)Li(x^[1/2]) - (1/3)Li(x^[1/3]) - (1/5)Li(x^[1/5]) + (1/6)Li(x^[1/6]) - +...

where in this sum, only square - free integers appear, and the sign is determined by the number of prime factors, minus if an odd number, or plus if an even number.

Empirically this approximation is indeed superior to Gauss's, but Riemann's argument for the error expression between this formula and pi(x), i.e. for his claimed theoretical accuracy of his approximation, is apparently what needs the Riemann hypothesis on just where the zeroes of the zeta function are located for a rigorous proof.

I learned all this, such as it is, just today, from reading riemann and then edwards. I do not pretend to have understood either, but, if even roughly correct, this is a lot more than i knew before. the moral is that one learns some useful facts really quickly by reading riemann, more than in a lifetime of reading others.

 

[jackmell]

To echo mathwonk and DonAntonio, stop and think about this for a second: why in the world does analysis (let alone complex analysis) tell you anything about number theory? The answer is of course not a priori obvious, and in fact remains quite mysterious to this day. If you approach analytic number theory with the motivation of answering this question then you might actually find yourself enjoying it.

It works because of jump-discontinuities, just like the integers. Consider the integral:

$$\mathop\oint\limits_C \frac{1}{z}dz$$

as a function of a unit circle moving up along the imaginary axis. The integral is zero until it just touches the origin, changes abruptly to $\pi i$, nothing in between, it's zero, zero, zero, then abruptly $\pi i$, then abruptly to $2\pi i$ as it engulfs the origin, then $-\pi i$, then back to zero. If contour integration over analytic functions did not have this peculiar property, then I suspect it would not work so well in Number Theory.

Also, in regards to Riemann's paper, he skipped 8 pages worth of explanations so it should have been 16.