First Hardy–Littlewood conjecture

Template:Short description Template:Infobox mathematical statement

In number theory, the first Hardy–Littlewood conjectureTemplate:Sfn 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 J. E. Littlewood in 1923.^[1]

Statement

Let $m_{1}, m_{2}, \dots, m_{k}$ be positive even integers such that the numbers of the sequence $P = (p, p + m_{1}, p + m_{2}, \dots, p + m_{k})$ do not form a complete residue class with respect to any prime and let $π_{P} (n)$ denote the number of primes $p$ less than $n$ so that $p + m_{1}, p + m_{2}, \dots, p + m_{k}$ are all prime. ThenTemplate:Sfn Template:Sfn

π_{P} (n) \sim C_{P} \int_{2}^{n} \frac{d t}{\log^{k + 1} t},

where

C_{P} = 2^{k} \prod_{\binom{q prime,}{q \geq 3}} \frac{1 - \frac{w (q; m_{1}, m_{2}, \dots, m_{k})}{q}}{{(1 - \frac{1}{q})}^{k + 1}}

is a product over odd primes and $w (q; m_{1}, m_{2}, \dots, m_{k})$ denotes the number of distinct residues of $0, m_{1}, m_{2}, \dots, m_{k}$ modulo $q$ .

The case $k = 1$ and $m_{1} = 2$ is related to the twin prime conjecture. Specifically if $π_{2} (n)$ denotes the number of twin primes less than n then

π_{2} (n) \sim C_{2} \int_{2}^{n} \frac{d t}{\log^{2} t},

where

C_{2} = 2 \prod_{\binom{q prime,}{q \geq 3}} (1 - \frac{1}{(q - 1)^{2}}) \approx 1.320323632 \dots

is the twin prime constant.Template:Sfn

Skewes' number

Template:Main article The Skewes' numbers for prime k-tuples are an extension of the definition of Skewes' number to prime k-tuples based on the first Hardy–Littlewood conjecture. The first prime p that violates the Hardy–Littlewood inequality for the k-tuple P, i.e., such that

π_{P} (p) > C_{P} {li}_{P} (p),

(if such a prime exists) is the Skewes number for P.Template:Sfn

Consequences

The conjecture has been shown to be inconsistent with the second Hardy–Littlewood conjecture.^[2]

Generalizations

The Bateman–Horn conjecture generalizes the first Hardy–Littlewood conjecture to polynomials of degree higher than 1.Template:Sfn

Notes

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References

Template:Prime number conjectures

[original-1] Template:Cite journal.

[2] Template:Cite journal

[1]

[2]

First Hardy–Littlewood conjecture

Contents

Statement

Skewes' number

Consequences

Generalizations

Notes

References

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First Hardy–Littlewood conjecture

Statement

Skewes' number

Consequences

Generalizations

Notes

References

Navigation menu

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