Analytic number theory

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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. Another major milestone in the subject is the prime number theorem.

Analytic number theory can be split up into two major parts. Multiplicative number theory deals with the distribution of the prime numbers, often applying Dirichlet series as generating functions. It is assumed that the methods will eventually apply to the general L-function, though that theory is still largely conjectural. Additive number theory has as typical problems Goldbach's conjecture and Waring's problem.

The development of the subject has a lot to do with the improvement of techniques. The circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of diophantine approximation are for auxiliary functions that aren't generating functions - their coefficients are constructed by use of a pigeonhole principle - and involve several complex variables. The fields of diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.

The biggest single technical change after 1950 has been the development of sieve methods as an auxiliary tool, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. Also much cited are uses of probabilistic number theory — forms of random distribution assertions on the primes, for example: these have not received any definitive shape. The extremal branch of combinatorial theory has in return been much influenced by the value placed in analytic number theory on (often separate) quantitative upper and lower bounds.

\, a_n \,

Contents

Some problems and results in analytic number theory

\,  p_n \,
\limsup_n ( p_n - p_{n-1}) \, = \infty \,  2+ N!,\,3 + N!,\, ...,\, N + N!
\liminf_n ( p_n - p_{n-1}) \,
\,  p_n \,
No one knows.
\, N \,

It took about 30 years for the mathematical community to digest Riemann's ideas and in the late 19th century, Hadamard, von Mangolt, and de la Vallee Poussin, made substantial progress in the field. In particular, they proved that if π(x) = { number of primes ≤ x } then

\lim_{x \to \infty} \frac{\pi(x)}{x/\log x} = 1
\, N\, \Re(s) = 1 \,

The Riemann zeta function

\, \sum_{n=1}^{\infty} \frac {1}{n^s} = \prod_p \frac {1}{1-p^{-s}} \,

Analysis and Number Theory

One may ask why exactly it is that analysis/calculus can be applied to number theory. One is "continuous" in nature and the other is "discrete" after all. Following Dirichlet's proof of the general theorem of primes in arithmetic progressions, mathematicians asked the exact same question. In fact, this was the motivation for developing a rigourous definition (and hence a rigorous theory) of the set of real numbers, R. At the time of Dirichlet's proof of his theorem, the notions of real number and (hence) the methods of analysis/calculus were based largely on physical/geometric intuition. It was thought somewhat disturbing that number theoretical conclusions were being deduced in a manner apparently reliant on such considerations, and it was thought desirable to find a number theoretical basis for these conclusions. This story has the following happy ending: It eventually turned out that there could be more rigourous definitions of real number, and that the (necessary) considerations involved in giving these definitions were the same as the considerations of elementary number theory: Induction, and addition and multiplication of arbitrary whole numbers. Therefore, we should not be particularly surprised at the application of analysis in number theory.

Paul Erdős

\, \sum \frac {1}{a_n} = \infty \,

Hardy, Littlewood

\, \Re(z) = 1/2 \,

The circle problem

\, \pi r^2 + E(r) \,

Notes

  1. ^ Page 7 of Apostol 1976
  2. ^ Page 1 of Davenport 2000
  3. ^ Page 7 of Apostol 1976
  4. ^ Page 56 of Tenenbaum 1995
  5. ^ Page 267 of Tenenbaum 1995

References

Further reading

On specialized aspects the following books have become especially well-known:

Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.

v • d • e Major topics in Number theory
Algebraic number theory • Analytic number theory • Geometric number theory • Computational number theory • Transcendental number theory
Numbers • Natural numbers • Prime numbers • Rational numbers • Irrational numbers • Algebraic numbers • Transcendental numbers • Arithmetic • Modular arithmetic • Arithmetic functions
Quadratic forms • L-functions • Diophantine equations • Diophantine approximation • Continued fractions
List of recreational number theory topics • List of number theory topics