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What are some of the big differences between analytic number theory and algebraic number theory?

Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much of analysis in analytic number theory.

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Err...the huge ammount of abstract algebra the latter requires as compared with the former? The first one also uses hefty ammounts of analysis, both real but specially complex. –  DonAntonio Jun 18 '12 at 16:36
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I guess my view is different. In classical analytic number theory, analysis seems to play the dominant role. –  André Nicolas Jun 18 '12 at 16:37
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Have you seen a proof of the prime number theorem? –  tomcuchta Jun 18 '12 at 16:43
    
@tomcuchta - not really. –  Victor Jun 18 '12 at 16:45
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It is not clear from your post how much of analytic, respectively algebraic number theory you have actually seen. So the only generic answer one can give at this point is: Why don't you go and actually learn the mathematics? Then you will be able to form your own opinion on differences and similarities. –  Alex B. Jun 18 '12 at 18:06

2 Answers 2

up vote 7 down vote accepted

The obvious answer is that Algebraic Number Theory uses techniques from algebra to answer number theory questions, while Analytic Number Theory uses techniques from analysis.

But that's a bit pedantic.

There is a certain degree to which analytic number theory is "about" the order properties of the natural numbers. For example, a lot of analytic number theory is trying to answer "how many examples of (some property) exists between $1$ and $N$?" The seminal example of this is "How many primes are between $1$ and $N$?" The answers to these are often estimations, and, in computing bounds on the errors, we are always using analysis.

Algebraic number theory is relatively unconcerned with this sort of question, and is often ill-equipped to answer it. Algebraic number theory can often be used to answer "Are there infinitely many?" but it often has a hard time dealing with bounds. It tends to be about bigger structural properties of the integers (and related number rings.)

As with any such generalizations, there are always exceptions.

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Here's a flavor of the kind of concepts that are introduced in the first three chapters of Tom Apostol's Introduction to Analytic Number Theory (Undergraduate Texts in Mathematics, Springer):

  1. Divisibility, greatest common divisors, prime numbers, Fundamental Theorem of Arithmetic, Euclidean algorithm. Series of reciprocals of the primes.

  2. Arithmetical Functions, Dirichlet product, Möbius inversion formula, Formal power series, Bell series of an arithmetical function, Derivatives of arithmetical functions, Selberg identity.

  3. Big oh notation, asymptotic equality of functions, averages of arithmetical functions.

Chapter 4 deals with the distribution of primes. Congruences are not introduced until Chapter 5; some results on finite abelian groups and their characters occupy Chapter 6. Their main purpose is to tackle Dirichlet's Theorem of primes in arithmetic progressions on Chapter 7. The book continues after that.

By contrast, William LeVeque's Fundamentals of Number Theory (Dover), which leans more towards the algebraic side, we have that Section 1.1 is titled What is number theory?, followed immediately by section 1.2, Algebraic properties of the set of integers. The first major chapter is Chapter 2, dealing with unique factorization and the GCD, much like Apostol (but not dealing at all with a series, whereas Apostol already has a series in that first chapter). Chapter 3 deals with congruences, Chapter 4 with primitive roots and the group of units modulo $m$, Chapter 5 with quadratic residues and quadratic reciprocity, and not until Chapter 6 are arithmetical functions introduced.

Serge Lang's Algebraic Number Theory (Graduate Texts in Mathematics, Springer) does not even speak about arithmetical functions. We go directly to unique factorization of ideals in Dedekind domains. Though it does have a part entitled Analytic Theory (Part 3, comprising chapters XIII through XVII), they are concerned with the zeta function, Tate's Thesis, the density of primes, and the Brauer-Siegel Theorem.

Questions like "What is the probability that two 'random' integers are relatively prime?" (Answer: $\frac{6}{\pi^2}$) "What is the average order of the divisor function?" (Answer, $$ \sum_{n\leq x} = x\log x + (2C-1)x + O(\sqrt{x})$$ where $C$ is Euler's constant); these are the province of Analytic Number Theory. Most of these questions cannot be reasonably answered (if at all) with the standard tools of algebraic number theory (Galois theory, extensions of $\mathbb{Q}$, rings of integers, etc.) Just like Lang's book does not even mention arithmetic functions, Apostol's does not even mention Galois.

Though both seek to answer questions about the properties of the positive integers, the kind of questions that Analytic Number Theory and Algebraic Number Theory ask have a distinct flavor, with the former concerned with "limiting" questions while the latter is not, and the latter being concerned with "structural" questions while the former not so much. And the kinds of tools that each reaches for is likewise different. Thus, while Algebraic Number Theory can tell you that there are infinitely many primes of each of the forms $4n+1$ and $4n+3$, and Algebraic Number Theory can even tell you that they each occur with density $\frac{1}{2}$ among all primes (via Cebotarev's Density Theorem), Algebraic Number Theory would have a hard time proving that "running totals" change leads infinitely often (that is, that there are infinitely many integers $N$s such that the number of positive primes less than $N$ that are congruent to $1$ mod $4$ is larger than the number of primes less than $N$ that are congruent to $3$ mod $4$, and that there are infinitely many $M$s such that the number of positive primes less than $M$ congruent to $3$ mod $4$ is larger than the number of positive primes less than $N$ congruent to $1$ mod $4$). In fact, I'm not aware of any "algebraic" proof of this fact.

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