Analytic number theory
The analytic number theory is a branch of number theory, which in turn is a branch of mathematics.
The analytic number theory used methods of analysis and theory of functions. Its content is mainly concerned with the determination of the number of all numbers below a given barrier, which have a certain property, as well as with the estimation of sums of number-theoretic functions.
Subdivisions and typical problems
Theory of Dirichlet series
To a sum
You want to examine, we consider the Dirichlet series f generated by the number theoretic function
Often the sum can be approximated as an integral over F (s ) can be expressed ( by an inverse Mellin transform), or its limit for x is obtained as the limit approaches infinity of F (s ) for s to 0 by a simulants. Therefore, the study of Dirichlet series and their generalizations (eg, the Hurwitz zeta function ) is a branch of number theory.
Multiplicative Number Theory
In particular, considering the case f = 1 and the associated Dirichlet series ( the Riemann zeta function ) leads to the prime number theorem that specifies the number of primes below a given limit. The investigation of the error term is an open problem, since the location of the zeros of the zeta function is unknown ( Riemann Hypothesis ). Similar methods are also applicable to other functions and provide conclusions about multiplicative the distribution of values (for example, on the frequency of abundant numbers).
Theory of Characters
Important multiplicative functions are the so-called characters; they are needed, if only numbers in certain residue classes counted or is to be summed over. So you can for example, demonstrate that one quarter of all prime numbers than last decimal place a 1, 3, 7 and 9 have, for details see Dirichlet's prime number theorem. Even Characters provides for the determination of the zeros of the associated Dirichlet series (L series), a major unsolved problem dar. (→ See Generalized Riemann conjecture ).
In addition, different sums of n-th complex roots of unity are examined: Character sums, especially Ramanujansummen. The theory of such sums is now considered as an independent branch.
Additive Number Theory
The additive number theory is concerned with the representation of numbers as sums. Oldest branch is the theory of partitions. Famous problems are the Waring problem ( representation of an integer as a sum of squares, cubes, etc.) and Goldbach's Conjecture ( every even number can be written as a sum of two primes? ). With the latter closely related to the presumption of twin primes is ( there infinitely many prime pairs with distance 2?).
Diophantine approximation and transcendental numbers
In addition, methods of analytic number theory are also used to prove the transcendence of numbers as the circle number, or Euler's number. Traditionally used is the field of Diophantine approximation: irrational numbers, which can be well approximated by rational numbers with small denominator ( Liouville number), form the oldest known class of transcendental numbers.
Applications
The classic questions of the area have not been made out of a practical need. More recently, results of analytic number theory play a role in the analysis of algorithms ( primality testing, factorization algorithms, random number generators ).