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.[1] 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.[1][2] 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[3] 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[4] — 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.

One of the recent breakthroughs in the field is Green's and Tao's proof of the existence of arbitrarily long arithmetic progressions in the primes.

Contents

[edit] Some problems and results in analytic number theory

One example of a well-known problem in analytic number theory is the Twin prime conjecture which asks whether there are infinitely many primes p such that p + 2 is prime. On the assumption of the Elliott-Halberstam conjecture it has been proven recently (by Daniel Goldston, János Pintz, Cem Yıldırım) that there are infinitely many primes p such that p + k is prime for some positive k less than 20.

The Prime Number Theorem is probably one of the most famous and interesting results in analytic number theory. Euclid showed that there are an infinite number of primes but it is very difficult to find an efficient method for determining whether or not a number is prime, especially a large number. Making this determination is known as primality testing. A related question one may hope to answer is whether or not the primes are distributed in some regular manner. Gauss, amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral

\, \int^N_2 \frac{1}{\log(t)} \, dt.

Bernhard Riemann, in 1859, used complex analysis and a special meromorphic function now known as the Riemann zeta function, ζ to derive an analytic expression for the number of primes less than or equal to a real number x. Remarkably, the main term in Riemann's formula was

\, \int^x_2 \frac{1}{\log(t)} \, dt,

confirming Gauss's conjecture. Riemann found that the manner in which the primes are distributed is closely related to the complex zeros of ζ.

It took about 30 years for the mathematical community to digest Riemann's ideas and in the late 19th century, Jacques Hadamard, Hans Carl Friedrich von Mangoldt, and Charles Jean de la Vallée-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.

This remarkable result is what is now known as the Prime Number Theorem. Loosely speaking, it states that given a large number \, N\,, the number of primes less than or equal to N is about N/log(N).

Analytic number theorists are often interested in the error of approximations such as these. In this case, the error is smaller than x/logx. It turns out that both of the first proofs[vague] of the prime number theorem heavily relied on the fact that ζ(s) ≠ 0 when \Re(s) = 1 \, and that the error can best be described if we know the location of all the complex zeros of ζ(s). In his 1859 paper[citation needed], Riemann conjectured that all the "non-trivial" zeros of ζ lie on the line \, \Re(s) = 1/2 \, but never provided a proof of this statement. This famous and long-standing conjecture is known as the Riemann Hypothesis and has many deep implications in number theory; in fact, many important theorems have been proved under the assumption that the hypothesis is true. For example under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is O(x^{1/2+\varepsilon}).

[edit] The Riemann zeta function

Main article: Riemann zeta function

Euler discovered that

\sum_{n=1}^\infty \frac {1}{n^s} = \prod_p \frac {1}{1-p^{-s}}\text{ for }s > 1. \,

Riemann considered this function for complex values of s and showed that this function can be extended to a meromorphic function on the entire plane with a simple pole at s = 1. This function is now known as the Riemann Zeta function and is denoted by ζ(s). There is a plethora of literature on this function and the function is a special case of the more general Dirichlet L-functions. Edwards' book, The Riemann Zeta Function is a good first source to study the function as Edwards goes over Riemann's original paper in depth and uses basic techniques learned in most first and second year graduate classes. Basic understanding of complex analysis and Fourier analysis are required for this reading.

[edit] 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. Leonard Euler must get credit for the first use of analytical arguments for the purpose of studying properties of integers, specifically by constructing generating power series[5]. Euler's proof of the infinity of prime numbers[6] makes use of the divergence of the zeta function [7] and the corresponding product over primes, which is named after him. This was the beginning of analytic number theory. [8] 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 rigorous definition (and hence a rigorous theory) of the set of real numbers, \mathbb{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 rigorous 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.

[edit] Hardy, Littlewood

In the early 20th century G.H.Hardy and Littlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis. In fact, in 1914, Hardy proved that there were infinitely many zeros of the zeta function on the critical line \, \Re(z) = 1/2 \, . This led to several theorems describing the density of the zeros on the critical line.

They also developed the circle method in order to study some problems in additive number theory like the Waring problem.

[edit] Paul Erdős

Paul Erdős was a great mathematician in the 20th century who is responsible for shaping much of the research in analytic number theory. He discovered many results in the field and also conjectured countless problems many of which remain unsolved to this day. The Tao-Green result on arithmetic progressions of primes is a partial solution to Erdős' conjecture that any sequence of positive integers such that \, \sum \frac {1}{a_n} = \infty \, contains arithmetic progressions of arbitrary length. Noam Elkies, a Harvard number theorist, writes that "mathematicians come in two types: theory builders and problem solvers, and analytic number theorists usually are from the problem solving camp." Paul Erdős was a very prolific problem solver. Many of his conjectures can be found in Guy's "Unsolved Problems in Number Theory."

[edit] Gauss' circle problem

Main article: Gauss circle problem

Given a circle centered about the origin in the plane with radius r, how many integer lattice points lie on or inside the circle? It is not hard to prove that the answer is \, \pi r^2 + E(r) \, , where \, E(r)/r^2 \, \to 0 \, as \, r \to \infty \,. Once again, we wish to bound the error term as precisely as possible.

As Gauss well knew, it is easy to show that E(r) = O(r). In general, an O(r) error term would be possible with the unit circle (or, more properly, the closed unit disk) replaced by the dilates of any bounded planar region with piecewise smooth boundary. Furthermore, replacing the unit circle by the unit square one sees that the difference between the area and the number of lattice points can in fact be as large as a linear function of r. Therefore getting an error bound of the form O(rδ) for some δ < 1 is a significant improvement. The first to attain this was Sierpinski in 1906, who got E(r) = O(r2 / 3). Circa 1915, Hardy and Landau each showed that one does not have E(r) = O(r1 / 2). Since then the goal has been to show that for each fixed ε > 0 there exists a real number C(ε) such that E(r) \leq C(\epsilon) r^{1/2 + \epsilon}.

In 1990 Huxley showed that E(r) = O(r47 / 63), which is the best published result. However, in February 2007 Cappell and Shaneson released a preprint which claims a full proof of the above (essentially) optimal bound on the error term. As of October 2008 the refereeing process on their paper is not yet complete.

[edit] Notes

  1. ^ a b Apostol 1976, p. 7.
  2. ^ Davenport 2000, p. 1.
  3. ^ Tenenbaum 1995, p. 56.
  4. ^ Tenenbaum 1995, p. 267.
  5. ^ http://en.wikipedia.org/wiki/Generating_function
  6. ^ http://en.wikipedia.org/wiki/Prime_number
  7. ^ http://en.wikipedia.org/wiki/Riemann_zeta_function
  8. ^ Iwaniec & Kowalski: Analytic Number Theory, AMS Colloquium Pub. Vol.53, 2004

[edit] References

[edit] Further reading

  • Ayoub, Introduction to the Analytic Theory of Numbers
  • H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I : Classical Theory
  • H. Iwaniec and E. Kowalski, Analytic Number Theory.
  • D. J. Newman, Analytic number theory, Springer, 1998

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
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