What Are Algebraic and Analytic Number Theory?

What is Analytic Number Theory

Analytic Number Theory is a branch of Mathematics that uses analytic techniques (for example - complex, Fourier) to attack problems in Number Theory.

The subject arguably starts from Riemann's idea on proving the Prime Number Theorem. The Prime number theorem states that the number of primes up to $x$ is roughly of the size $$\dfrac{x}{\log x}$$ as $x$ growths to infinity. It was Gauss who first proposed the Prime Number Theorem (without a proof), but no one knows how to prove this for a long time. Riemann had the insight that this is related to the behaviour of the zeta function via complex analysis. It was eventually rigorously proved this way by Hadamard and de la Poussin.

Modern analytic number theory is a very broad subject, but the general theme remains to use methods from analysis to attack problems in number theory. Some broad, modern directions include understanding Riemann's zeta function, or L-functions in general; understanding distribution of prime numbers, where sieve theory has been prominent; understanding multiplicative functions, or additive problems in number theory.

What is Algebraic Number Theory

Algebraic number theory concerns with algebraic structures related to algebraic integers. The question that naturally comes is: what is algebraic integer? A number $x$ is an algebraic integer if it is a root of a polynomial which has coefficients in $\mathbb Z$. Mostly we consider a monic polynomial in such cases. A polynomial is monic if it has leading coefficient $1$. So, $x^2+x-1$ is monic but $2x+3$ is not.

What does the definition really mean? In fact, what associates most with Algebraic number theory is Abstract Algebra. You can think of this as a generalization/abstraction of the objects or structures you have encountered so far based on some properties or operations. Here, an operation can mean anything, from multiplication or addition to rotation, that is, literally anything. Some important topics that belong to this section are Groups, Rings, Fields, Domain, Ideal. The basic of all these structures is they are all sets. The theories related to a particular structure is developed based on set properties, operations and the axioms assumed.