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I try to explaine my problem and I hope do not disturb or annoy;

I know that number theory is very vast but essentially it is divided into two parts:

analytic number theory and algebraic number theory;

My questions are: what are the main tool used for working on it, the main topic that one can study and the main active reaserch areas in this two sectors?

what are their applications to study the property of the integers?

what is in general number theory today?

Thanks to all for your valuable answers!

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closed as too broad by anomaly, Mathmo123, Henrik, Joffan, Greg Martin Jun 8 '16 at 22:54

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Nowadays, number theory is largely popularized by cryptography and its application in said area (check out RSA). $\endgroup$ – Victor Jun 8 '16 at 21:46
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    $\begingroup$ @Victor: Although cryptography is a useful application of number theory, it's certainly not the most important, productive, or significant area of the field. $\endgroup$ – anomaly Jun 8 '16 at 21:48
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    $\begingroup$ Number theory isn't a collection of tools, and there's a huge variety of topics and active research areas within it. In addition to analytic number theory and algebraic number theory, I'd also add arithmetic geometry as a separate part of the field. $\endgroup$ – anomaly Jun 8 '16 at 21:50
  • $\begingroup$ You'd be tight stretched to fit a summary of modern number theory into a chapter of a book, let alone a stack exchange answer. This is impossibly broad! Is there any way you can narrow down your question? $\endgroup$ – Mathmo123 Jun 8 '16 at 21:56
  • $\begingroup$ Take a look at this link: numbertheory.org/ntw/number_theory.html $\endgroup$ – user72870 Jun 8 '16 at 22:06
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It's not really as black and white as that. A lot of Algebraic Geometry is studied in Number Theory to look at Elliptic Curves. Erdös even used probabilistic techniques to study the primes back in the 30s by essentially looking at the primes as random variables. Elementary Number Theory is also the study of integers without the use of Complex Analysis (as in Analytic NT) or algebraic structures (As in Algebraic NT).

Algebraic Number Theory uses techniques from abstract algebra to study the properties of the integers ($\Bbb Z$) and the rationals $\Bbb Q$, and extensions thereof. Other structures that are similar (for instance the Gaussian integers $\Bbb Z[i]$) are also studied to see HOW they are similar and why.

Analytic Number Theory uses techniques from Complex Analysis to infer results about the integers (strangely, since Complex Analytic objects tend to be continuous in nature).

There is a lot of overlap though; the theory of modular forms and elliptic curves merge in and out of either of the two fields.

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    $\begingroup$ maybe you could mention that complex analytic functions are not so continuous when looking at their zeros/poles and more generally their series/product expansions. I think this explains mostly why complex analytic functions are so useful in number theory $\endgroup$ – reuns Jun 8 '16 at 22:15

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