# Analytical number theory

The analytic number theory is a branch of number theory , which in turn is a branch of mathematics is.

Analytical number theory uses methods from analysis and function theory . In terms of content, it is mainly concerned with the determination of the number of all numbers below a given limit that have a certain property, as well as with the estimation of sums of number theoretic functions .

## Sub-areas and typical problems

### Theory of the Dirichlet series

To a sum

${\ displaystyle \ sum _ {n \ leq x} f (n)}$,

which one would like to investigate, one considers the Dirichlet series generated by the number theoretic function${\ displaystyle f}$

${\ displaystyle F (s): = \ sum _ {n = 1} ^ {\ infty} f (n) n ^ {- s}}$.

Often the sum can be expressed approximately as an integral over (by an inverse Mellin transformation ), or its limit value for towards infinity is obtained as a limit value for towards 0 through a deaf set . Therefore, the investigation of Dirichlet series and their generalizations (e.g. Hurwitz's zeta function ) forms a branch of number theory. ${\ displaystyle F (s)}$${\ displaystyle x}$${\ displaystyle F (s)}$${\ displaystyle s}$

### Multiplicative number theory

In particular, considering the case and the corresponding Dirichlet series (the Riemann zeta function ) leads to the prime number theorem , which indicates the number of prime numbers below a given bound. The investigation of the error term is an open problem, since the position of the zeros of the zeta function is unknown ( Riemann hypothesis ). Similar methods can also be used for other multiplicative functions and provide information about their value distribution (for example about the frequency of abundant numbers ). ${\ displaystyle f = 1}$

### Theory of characters

Important multiplicative functions are the so-called characters ; they are required if only numbers in certain remainder classes are to be counted or summed up. For example, one can prove that a quarter of all prime numbers each have a 1, 3, 7 or 9 as the last decimal place, for details see the Dirichletscher Prime Number Theorem . The determination of the zeros of the associated Dirichlet series (L series) is also a big unsolved problem for characters. (→ See generalized Riemann assumption ).

In addition, different sums of -th, complex roots of unity are examined: character sums , especially Ramanujan sums . The theory of such sums is now viewed as an independent sub-area. ${\ displaystyle n}$

Additive number theory deals with the representation of numbers as sums. The oldest branch is the theory of partitions . Famous problems are the Waring problem (representation of a whole number as a sum of squares, cubes etc.) and the Goldbach conjecture (can every even number be written as the sum of two prime numbers?). The conjecture about prime twins is closely related to the latter (is there an infinite number of prime number pairs with a distance of 2?).

### Diophantine approximation and transcendent numbers

In addition, methods of analytical number theory also serve to prove the transcendence of numbers such as the circular number${\ displaystyle \ pi}$ or Euler's number${\ displaystyle e}$ . The area of ​​Diophantine approximation is traditionally related: irrational numbers, which can be approximated well by rational numbers with a small denominator ( Liouville number ), form the oldest known class of transcendent numbers.

## Applications

The classic questions of the field have not been asked out of a practical need. More recently, the results of analytical number theory have played a role in the analysis of algorithms ( primality tests , factoring algorithms , random number generators ).