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 second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the

first Hardy–Littlewood conjecture In number theory, the first Hardy–Littlewood conjecture states the asymptotic formula for the number of prime ''k''-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Eden ...

, the second Hardy–Littlewood conjecture was proposed by

G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

and

John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...

in 1923..

Statement

The conjecture states that

\pi(x+y) \leq \pi(x) + \pi(y)

for integers , where denotes 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 ...

, giving the number of prime numbers up to and including .

Connection to the first Hardy–Littlewood conjecture

The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from to is always less than or equal to the number of primes from 1 to . This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime -tuples, and the first violation is expected to likely occur for very large values of . For example, an admissible ''k''-tuple (or prime constellation) of 447 primes can be found in an interval of integers, while . If the first Hardy–Littlewood conjecture holds, then the first such -tuple is expected for ' greater than but less than .

References

External links

* * Analytic number theory Conjectures about prime numbers Unsolved problems in number theory {{numtheory-stub