Analytic Number Theory

Analytic Number Theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. 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, divided more by the type of problems they attempt to solve than fundamental differences in technique. Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. Additive number theory is concerned with the additive structure of the integers, such as Goldbach's conjecture that every even number greater than 2 is the sum of two primes. One of the main results in additive number theory is the solution to Waring's problem.

Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, 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 technical change after 1950 has been the development of sieve methods as a tool, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses tools from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.

Problems and Results in Analytic Number Theory

The great theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate.

Multiplicative 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. A related but easier problem is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. 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.

In 1859 Bernhard Riemann used complex analysis and a special meromorphic function now known as 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 exactly the above integral, lending substantial weight to Gauss's conjecture. Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function. Using Riemann's ideas and by getting more information on the zeros of the zeta function, Jacques Hadamard and Charles Jean de la Vallée-Poussin managed to complete the proof of Gauss's conjecture. In particular, they proved that if

\pi(x) = (\text{number of primes }\leq 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. It is a central result in analytic number theory. 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).

More generally, the same question can be asked about the number of primes in any arithmetic progression a+nq for any integer n. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with a and q coprime contains infinitely many primes. The prime number theorem can be generalised to this problem; letting

then if a and q are coprime,

\lim_{x \to \infty} \frac{\pi(x,a,q)\phi(q)}{x/\log x} = 1.

There are also many deep and wide ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as theTwin 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 psuch that p + k is prime for some positive even k less than 16.

Additive Number Theory

One of the most important problems in additive number theory is Waring's problem, which asks whether it is possible, for any k ≥ 2, to write any positive integer as the sum of a bounded number of kth powers,

n=x_1^k+\cdots+x_\ell^k. \,

The case for squares, k = 2, was answered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved by Hilbert in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood. These techniques are known as the circle method, and give explicit upper bounds for the function G(k), the smallest number of kth powers needed, such as Vinogradov's bound

G(k)\leq k(3\log k+11). \,

Diophantine Problems

Diophantine problems are concerned with integer solutions to polynomial equations, and especially how many you can expect to find within a given range.

One of the most important examples is the Gauss circle problem, which asks for integers points (x y) which satisfy

x^2+y^2\leq r^2.

In geometrical terms, given a circle centered about the origin in the plane with radius r, the problem asks 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 \,. Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term E(r).

It was shown by Gauss 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, the error term for the general problem can be as large as a linear function of r. Therefore getting an error bound of the form O(r^{\delta}) for some \delta < 1 in the case of the circle is a significant improvement. The first to attain this was Sierpiński in 1906, who showed  E(r) = O(r^{2/3}). In 1915, Hardy and Landau each showed that one does not have E(r) = O(r^{1/2}). Since then the goal has been to show that for each fixed \epsilon > 0 there exists a real number C(\epsilon) such that E(r) \leq C(\epsilon) r^{1/2 + \epsilon}.

In 2000 Huxley showed that E(r) = O(r^{131/208}), which is the best published result.

Methods of Analytic Number Theory

Dirichlet Series

One of the most useful tools in multiplicative number theory are Dirichlet series, which are functions of a complex variable defined by an infinite series

f(s)=\sum_{n=1}^\infty a_nn^{-s}.

Depending on the choice of coefficients a_n, this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identity

\left(\sum_{n=1}^\infty a_nn^{-s}\right)\left(\sum_{n=1}^\infty b_nn^{-s}\right)=\sum_{n=1}^\infty\left(\sum_{k\ell=n}a_kb_\ell\right)n^{-s};

hence the coefficients of the product of two Dirichlet series are the multiplicative convolutions of the original coefficients. Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.

Riemann Zeta Function

Euler showed that the fundamental theorem of arithmetic implies that

 \sum_{n=1}^\infty \frac {1}{n^s} = \prod_p^\infty \frac {1}{1-p^{-s}}\text{ for }s > 1\,\,\ (p \text{ is prime number)} \,

Euler's proof of the infinity of prime numbers makes use of the divergence of the term at the left hand side for s = 1 (the so-called harmonic series), a purely analytic result. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructing generating power series. This was the beginning of analytic number theory.

Later, 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.

Analytic number theorists are often interested in the error of approximations such as the prime number theorem. In this case, the error is smaller than x/log x. Riemann's formula for π(x) shows that the error term in this approximation can be expressed in terms of the zeros of the zeta function. In his 1859 paper, 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}).

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.

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