Question

Difference between algebraic and analytic number theory

Answer 1

The 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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