In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, the Elliott–Halberstam conjecture is a conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 about the distribution of prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s in arithmetic progression

Arithmetic progression

In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...

s. It has many applications in sieve theory

Sieve theory

Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

. It is named for Peter D. T. A. Elliott and Heini Halberstam

Heini Halberstam

Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

.

To state the conjecture requires some notation. Let denote the number of primes less than or equal to x. If q is a positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 and a is coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to q, we let π(x;q,a), denote the number of primes less than or equal to x which are equal to a modulo q. Dirichlet's theorem on primes in arithmetic progressions then tells us
that


when a is coprime to q and is Euler's totient function

Euler's totient function

In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...

. If we then define the error function


where the max is taken over all a coprime to q, then the Elliott–Halberstam conjecture is the assertion that
for every θ < 1 and A > 0 there exists a constant C > 0 such that


for all x > 2.

This conjecture was proven for all θ < 1/2 by Enrico Bombieri

Enrico Bombieri

Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...

 and A. I. Vinogradov (the Bombieri–Vinogradov theorem

Bombieri–Vinogradov theorem

In mathematics, the Bombieri–Vinogradov theorem is a major result of analytic number theory, obtained in the mid-1960s. The first result of this kind was obtained by Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri-Vinogradov theorem is named...

, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the generalized Riemann hypothesis

Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...

. It is known that the conjecture fails at the endpoint θ = 1.

The Elliott–Halberstam conjecture has several consequences. One striking one is the recent result announced by Dan Goldston, János Pintz

János Pintz

János Pintz is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences.-Mathematical results:...

, and Cem Yıldırım

Cem Yildirim

Cem Yalçın Yıldırım is a Turkish mathematician who specializes in number theory. He obtained his PhD from the University of Toronto in 1990. His advisor was John Friedlander...

, which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16.