In mathematics, analytic number theory is a branch of number theory, which uses mathematical analysis to solve the problem of number theory. [1] It is generally believed that it started with Dirichlet's introduction of Dirichlet L- function, and the first proof of Dirichlet's theorem in arithmetic process is given. [2][ 1] Another major milestone of this discipline is the prime number theorem.
Analytic number theory can be divided into two main parts. Multiplication number theory studies the distribution of prime numbers, and Dirichlet series is often used as the generating function. It is assumed that these methods will eventually apply to general L- functions, although the theory is still largely speculative. The typical problems of additive number theory are Goldbach conjecture and Willing problem.
The development of disciplines is closely related to the progress of technology. Hardy and Littlewood's circle method is considered to be suitable for power series near the unit circle on the complex plane; It is now considered to be a finite exponential sum (that is, on the unit circle, but the power series is truncated). Diophantine approximation needs auxiliary functions, which are not generating functions-their coefficients are constructed by using pigeon hole principle-and involve several complex variables. The field of Diophantine approximation and transcendence theory has been expanded, so that these techniques have been applied to modal conjecture.
One of the biggest technical changes after 1950 is the development of screening method [3] as an auxiliary tool, especially in multiplication. These are combined in nature and vary greatly. Also widely cited is the use of probability number theory [4]-the form of assertion of random distribution on prime numbers, for example, these do not get any definite shape. The extreme branch of combinatorial theory is in turn influenced by the quantitative upper and lower limits (usually separated) in analytic number theory.
Ben Green and Tao Zhexuan proved one of the most profound and important theorems in analytic number theory in 2004. Using analytical methods, they proved that there is an arbitrary length of arithmetic series for prime numbers. This is part of the solution to Paul Erd? S conjecture: Any divergent positive integer sequence contains arithmetic series of arbitrary length.
Some problems and results in analytic number theory
1. Let's represent the nth prime number. What is and?
. To see this, let n be any big positive integer.
So the number is n? 1 continuous compound integer, and because n can be chosen to be arbitrarily large, this proves the result.
Is unknown. Although the estimated value is 2. This is an expression of the famous twin prime conjecture.
2. Let pn represent the nth prime number. This series
Convergence? Nobody knows.
3. The prime number theorem is probably one of the most famous and interesting results in analytic number theory. Mathematicians have been trying to understand prime numbers for hundreds of years. Euclid showed us that there are infinitely many prime numbers, but it is difficult to find an effective method to determine whether a number is a prime number, especially a large number. Wilson's theorem is such a result, but it is still very inefficient. For centuries, mathematicians have been trying to find a model to describe all prime numbers, but without much success. Continuing, the next question may be whether the prime numbers are distributed in some regular way. Gauss and others speculate that the number of prime numbers less than or equal to the large number n is close to the integral value.
Without the help of a computer, he calculated a very large list of prime numbers and guessed the result. In 1859, Bernhard Riemann used complex analysis and a very special function, Riemann zeta function, to derive the analytical expression of prime numbers less than or equal to real number X. It is worth noting that the main term in Riemann formula is
Confirmed gauss's guess. Riemann formula is not accurate, but he found that the distribution of prime numbers is closely related to the complex zero of a special meromorphic function, Riemann zeta function zeta (s). Therefore, a new number theory method was born.
It took about 30 years for mathematics to digest Riemann's thoughts. In the late19th century, Hadamard, von Mangolt and de la Vallee Poussin made substantial progress in this field. In particular, they proved that if π(x) = {prime number ≤ x} then
This remarkable result, called the prime number theorem, says that given a large number, the number of prime numbers 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 given by the prime number theorem is less than x/logx. But the next question is: How big can it be? It has been proved that the two initial proofs of the prime number theorem depend heavily on the fact that when ζ(s) is ≠ 0, and if we know the positions of all complex zeros of ζ(s), we can best describe the error. In his paper 1859, Riemann conjectured that all the "nontrivial" zeros of ζ were in a straight line, but he did not prove this statement. This conjecture, called Riemann hypothesis, is considered as the most important unsolved problem in mathematics. Riemann hypothesis is very important because it has many profound meanings in number theory; If it is true, then we can prove many theorems in number theory and gain a better understanding of prime numbers. In fact, assuming that the hypothesis is true, many important theorems have been proved. For example, under the assumption of Riemann hypothesis, the error term in the prime number theorem is.
[Edit] Riemannian Zeta function
Euler discovery
Riemann considers this function of complex value of S, and shows that this function can be extended to meromorphic functions on the whole plane with a simple pole at s = 1 This function is now called Riemann zeta function, and is denoted by zeta (s). There are a lot of literatures about this function, which is a special case of the more general Dirichlet L-function. Edwards' book "Riemann Zeta Function" is a good first-hand material to study this function, because Edwards deeply read Riemann's original paper and used the basic skills learned in most first-year and second-year graduate classes. Reading this article requires a basic understanding of complex analysis and Fourier analysis.
[Editor] Analysis and Number Theory
Some people may ask why analysis/calculus can be applied to number theory. One is "continuous" in essence and the other is "discrete" after all. After Dirichlet proved the general theorem of prime numbers in arithmetic series, mathematicians put forward exactly the same problem. In fact, this is the motivation to develop a strict definition (and therefore a strict theory) of the real number set R. When Dirichlet proved his theorem, the concept of real number and the method of analysis/calculus were mainly based on physical/geometric intuition. People think it is disturbing that the conclusions of number theory are derived in a way that obviously depends on this consideration, and people think it is desirable to find the basis of number theory for these conclusions. The story has the following happy ending: It turns out that real numbers may have stricter definitions, and the (necessary) considerations involved in giving these definitions are the same as those of elementary number theory: induction, addition and multiplication of arbitrary integers. Therefore, we should not be particularly surprised by the application of analysis in number theory.
[Editor] Littlewood Hardy
In the early 20th century, in order to prove Riemann hypothesis, G H Hardy and Littlewood proved many results about zeta function. In fact, in 19 14, Hardy proved that the zeta function has infinite zeros on the critical line. This leads to several theorems describing the zero density on the critical line.
They also developed the circle method in order to study some problems in the theory of additive numbers, such as the Waring problem.
[Editor] Paul Elder? s
Paul Roeder? S is a great mathematician in the 20th century, and he is responsible for the formation of most research on analytic number theory. He found many results in this field and speculated on countless problems, many of which have not been solved yet. Is the Tao-Green result of prime arithmetic series a partial solution of Erd? Guess that any positive integer sequence contains arithmetic series of arbitrary length. Noam Elkis, a Harvard number theorist, wrote: "There are two types of mathematicians: theoretical builders and problem solvers. Analytic number theorists usually come from the problem-solving camp." Paul Roeder? S is a very prolific problem solver. Many of his conjectures can be found in Guy's "Unsolved Problems in Number Theory".
[Editor] Gauss circle problem
Given a circle whose center is on a plane with radius r, how many integer grids are located on or within the circle? It is not difficult to prove that the answer is, where is as? Similarly, we want to define the error term as accurately as possible.
As Gauss knows, it is easy to prove that E(r) = O(r). Generally speaking, it is possible to replace the unit circle (or more appropriately, the closed unit circle) with the expansion of any bounded plane region with piecewise smooth boundaries, and the O(r) error term is possible. In addition, by replacing the unit circle with the unit square, we can see that the difference between the area and the number of lattice points can actually be as large as a linear function of R. Therefore, for some δ <: 1 is a remarkable improvement. The first one to achieve this is Sierpinski in 1906, and he gets E(r) = O(r2/3). About 19 15, Hardy and Landau each proved a nonexistent E(r) = O(r 1/2). Since then, our goal is to prove that for each fixed ε >; There is a real number C(ε) that makes.
In 1990, Huxley proved that E(r) = O(r47/63), which is the best published result. However, in February 2007, Cappell and Shaneson published a preprint, claiming that the optimal bounds of the above (essential) error terms were completely proved. As of 2008 10, the review process of their papers has not been completed.
[edit] comments
Page 7 of Acts, 1976
Davenport 2000 Page 1
Tenenbaum 1995 Page 56
Tenenbaum 1995 p. 267.
[edit] reference
Tom Apostol (1976), Introduction to Analytic Number Theory, textbook for undergraduate students in mathematics, new york-Heidelberg: springer Press, MR0434929, ISBN 978-0-387-90 163-3.
Davenport, Harold (2000), Multiplication Theory, Mathematics Postgraduate Textbook, 74 (Third Revision). ), new york: springer Publishing House, MR 1790423, ISBN 978-0-387-95097-6.
Tenenbaum, Gérald (1995), Introduction to Analytic and Probability Number Theory, Cambridge Advanced Mathematics Research, 46, Cambridge University Press, ISBN 0-521-41261-7.
[Edit] Extended reading
Ayub, Introduction to Digital Analytic Theory.
H.L. Montgomery and R.C. Vaughan, Multiplication Theory I: Classical Theory
H.Iwaniec and E. Kowalski, analytic number theory.
D Newman, Analytic Number Theory, springer, 1998.
Professionally, the following books have become particularly famous:
E.C. Titchmarsh, Theory of Riemann zeta Function, 2nd Edition.
H. Halberstam and richert, screening method; And R.C. Vaughn's Hardy-Littlewood method, second. edn。
Some topics have not been written in any depth. Some examples are: (1) Montgomery's pairing-related conjecture and the work started from it, (Goldston, Pintz and Yilidrim's new results about the small interval between prime numbers, and (3) Green-Tao theorem shows that there is an arbitrary length of arithmetic series for prime numbers.