In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the Elliott–Halberstam conjecture is a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
about the distribution of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s in
arithmetic progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
s. It has many applications in
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 prototypical example of a sifted set is the set of prime numbers up to some prescribed limi ...
. It is named for
Peter D. T. A. Elliott and
Heini Halberstam
Heini Halberstam (11 September 1926 – 25 January 2014) was a Czech-born British mathematician, working in the field of analytic number theory. He is remembered in part for the Elliott–Halberstam conjecture from 1968.
Life and career
Halber ...
, who stated a specific version of the conjecture in 1968.
One version of the conjecture is as follows, and stating it requires some notation. Let
, the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal ...
, denote the number of primes less than or equal to
. If
is a
positive integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
and
is
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
to
, we let
denote the number of primes less than or equal to
which are equal to
modulo
.
Dirichlet's theorem on primes in arithmetic progressions then tells us
that
:
where
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
. If we then define the error function
:
where the max is taken over all
coprime to
, then the Elliott–Halberstam conjecture is the assertion that
for every
and
there exists a constant
such that
:
for all
.
This conjecture was proven for all
by
Enrico Bombieri
Enrico Bombieri (born 26 November 1940) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently professor emeritus in the School of Mathematics ...
and
A. I. Vinogradov (the
Bombieri–Vinogradov theorem
In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over ...
, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the
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, whi ...
. It is known that the conjecture fails at the endpoint
. In 1986, Bombieri, Friedlander and Iwaniec generalized the Elliott-Halberstam conjecture, using
Dirichlet convolution
In mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet.
Definition
If f , g : \mathbb\to\mathbb ...
of arithmetic functions related to the
von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mang ...
.
The Elliott–Halberstam conjecture has several consequences. A striking one is the result announced by
Dan Goldston,
János Pintz, and
Cem Yıldırım, which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013,
James Maynard showed that subject to the Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12. In August 2014,
Polymath
A polymath or polyhistor is an individual whose knowledge spans many different subjects, known to draw on complex bodies of knowledge to solve specific problems. Polymaths often prefer a specific context in which to explain their knowledge, ...
group showed that subject to the
generalized Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6.
Without assuming any form of the conjecture, the lowest proven bound is 246.
Original conjecture
The original Elliott-Halberstam conjecture is not clearly stated in their paper,
but can be inferred there from (1) page 59 and the comment above the Theorem on page 62. It says that
:
provided