Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve number-theoretical problems. [Page 7 of Apostol 1976] 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. [Page 1 of Davenport 2000] [Page 7 of Apostol 1976] 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" [Page 56 of Tenenbaum 1995] 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" [Page 267 of Tenenbaum 1995] — 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 deepest and most important theorems in analytic number theory has been proven by Ben Green and Terence Tao in 2004. Using analytic methods, they proved that there exists arbitraly long arithmetic progressions of prime numbers. This is a partial solution to Paul Erdős' conjecture that any sequence of positive integers $,\; a\_n\; ,$ such that $,\; sum\_n\; frac\; \{1\}\{a\_n\}$ diverges, contains arithmetic progressions of arbitrary length.

Some problems and results in analytic number theory

1. Let $,\; p\_n\; ,$ denote the "n"th prime. What are $limsup\_n\; (\; p\_n\; -\; p\_\{n-1\})\; ,$ and $liminf\_n\; (\; p\_n\; -\; p\_\{n-1\})\; ,$?

:: $limsup\_n\; (\; p\_n\; -\; p\_\{n-1\})\; ,\; =\; infty\; ,$. To see this let "N" be any large positive integer.:Then the numbers $2+\; N!,,3\; +\; N!,,\; dots,,\; N\; +\; N!$ are "N" − 1 consecutive composite integers and since "N" can be chosen to be arbitrarily large this proves the result.

:: $liminf\_n\; (\; p\_n\; -\; p\_\{n-1\})\; ,$

: is unknown. Although it is conjectured that the value is 2. This is one way of stating the famous twin prime conjecture.

2. Let "p"_"n" denote the "n"th prime. Does the series

:: $sum\_n\; (-1)^n\; frac\; \{n\}\{p\_n\}$

: converge? No one knows.

3. The Prime Number Theorem is probably one of the most famous and interesting results in analytic number theory. For hundreds of years mathematicians have been trying to understand prime numbers. Euclid has shown us 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. Wilson's theorem is one such result but it is still very inefficient. Mathematicians have tried for centuries to find a pattern that describes all the prime numbers without much success. Moving on, the next question one may hope to answer is whether or not the primes are distributed in some regular manner. Gauss, among others, conjectured that the number of primes less than or equal to a large number "N" is close to the value of the integral

:: $,\; int^N\_2\; frac\{1\}\{log(t)\}\; ,\; dt.$

: Without the aid of a computer he computed very large lists of primes and guessed this result. Bernhard Riemann, in 1859, used complex analysis and a very special function, the Riemann zeta function, 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 guess. Riemann's formula was not exact but he found that the manner in which the primes are distributed is closely related to the complex zeros of a special meromorphic function, the Riemann Zeta function "ζ"("s"). Hence, a new approach to number theory was born.

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

:$lim\_\{x\; o\; infty\}\; frac\{pi(x)\}\{x/log\; x\}\; =\; 1.$

This remarkable result, known as the Prime Number Theorem, says that given a large number $,\; N,$, then 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 such results. The error tends to 0 as "N" tends to infinity but the (next) question is: how fast or how slow?. In other words, is there a formula that describes the error? It turns out that both of the first proofs 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, Riemann conjectured that all the "non-trivial" zeros of ζ lie on the line $,\; Re(s)\; =\; 1/2\; ,$ but he did not prove this statement. This conjecture is known as the Riemann Hypothesis and is believed to be the most important unsolved problems in mathematics. The Riemann Hypothesis is important because it has many deep implications in number theory; if its true then we can prove many theorems in number theory and gain a better understanding of prime numbers. In fact, many important theorems have been proved assuming the hypothesis is true. There are also proofs whose structure involves splitting into two cases, according to whether the Riemann hypothesis is true or false. A resolution of the Riemann hypothesis would automatically make this unnecessary.

The Riemann zeta function

Euler discovered that

: $sum\_\{n=1\}^infty\; frac\; \{1\}\{n^s\}\; =\; prod\_p\; frac\; \{1\}\{1-p^\{-s\; ext\{\; for\; \}s\; >\; 1.\; ,$

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 know 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 Diriclet 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.

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 rigorous 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 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.

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."

Hardy, Littlewood

In the early 20th century 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 theoremsdescribing the density of the zeros on the critical line. For example, they proved that a positive proportion of the zeros lie on the line and this is one the strongest theoremsthat supports the validity of the Riemann Hypothesis. They created several techniques, such as the Hardy - Littlewood "circle method", that are still used by today by experts in the field.

The circle problem

One interesting problem in number theory is Gaub's lattice problem now known simply as the circle problem: Given a circle centered about the origin in the plane with raduis, r, how many integer lattice points are in this circle? It is not hard to prove that the answer is $,\; pi\; r^2\; +\; E(r)\; ,$ where $,\; E(r)\; ,\; o\; 0\; ,$ as $,\; r\; o\; infty\; ,$. Once again, we wish to understand the asymptotics of the error term. It is conjectured that $,\; E(r)\; =\; O(r^\{1/4\})\; ,$ and many techniques have been developed over the years to get the desired result but the problem still remains open to this day. Note that one may ask the corresponding question for spheres, ellipsoids, etc.

Notes

References

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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:

* E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd.edn.
* H. Halberstam and H. -E. Richert, Sieve Methods; and R. C. Vaughan, The Hardy-Littlewood method, 2nd. edn.

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.