Battle of Round Mountain

In number theory, the first Hardy–Littlewood conjecture^[1] states the asymptotic formula for the numer 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 Edensor Littlewood in 1923.^[2]

Statement

Let $m_{1},m_{2},\ldots ,m_{k}$ be positive even integers such that the numbers of the sequence $P=(p,p+m_{1},p+m_{2},\ldots ,p+m_{k})$ do not form a complete residue class with respect to any prime and let $\pi _{P}(n)$ denote the number of primes $p$ less than $n$ st. $p+m_{1},p+m_{2},\ldots ,p+m_{k}$ are all prime. Then^[1]^[3]

\pi _{P}(n)\sim C_{P}\int _{2}^{n}{\frac {dt}{\log ^{k+1}t}},

where

C_{P}=2^{k}\prod _{q{\text{ prime,}} \atop q\geq 3}{\frac {1-{\frac {w(q;m_{1},m_{2},\ldots ,m_{k})}{q}}}{\left(1-{\frac {1}{q}}\right)^{k+1}}}

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

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

\pi _{2}(n)\sim C_{2}\int _{2}^{n}{\frac {dt}{\log ^{2}t}},

where

C_{2}=2\prod _{\textstyle {q{\text{ prime,}} \atop q\geq 3}}\left(1-{\frac {1}{(q-1)^{2}}}\right)\approx 1.320323632\ldots

is the twin prime constant.^[3]

Skewes' number

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

\pi _{P}(p)>C_{P}\operatorname {li} _{P}(p),

(if such a prime exists) is the Skewes number for P.^[3]

Consequences

The conjecture has been shown to be inconsistent with the second Hardy–Littlewood conjecture.^[4]

Generalizations

The Bateman–Horn conjecture generalizes the first Hardy–Littlewood conjecture to polynomials of degree higher than 1.^[1]

Notes

^ ^a ^b ^c Aletheia-Zomlefer, Fukshansky & Garcia 2020.
^ Hardy, G. H.; Littlewood, J. E. (1923). "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes". Acta Math. 44 (44): 1–70. doi:10.1007/BF02403921..
^ ^a ^b ^c Tóth 2019.
^ Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438. doi:10.1090/S0002-9904-1974-13434-8.

References

Aletheia-Zomlefer, Soren Laing; Fukshansky, Lenny; Garcia, Stephan Ramon (2020). "The Bateman–Horn conjecture: Heuristic, history, and applications". Expositiones Mathematicae. 38 (4): 430–479. doi:10.1016/j.exmath.2019.04.005. ISSN 0723-0869.
Tóth, László (January 2019). "On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood". Computational Methods in Science and Technology. 25: 143–138. arXiv:1910.02636. doi:10.12921/cmst.2019.0000033.

[FOOTNOTEAletheia-ZomleferFukshanskyGarcia2020-1] Aletheia-Zomlefer, Fukshansky & Garcia 2020.

[original-2] Hardy, G. H.; Littlewood, J. E. (1923). "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes". Acta Math. 44 (44): 1–70. doi:10.1007/BF02403921..

[FOOTNOTETóth2019-3] Tóth 2019.

[4] Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438. doi:10.1090/S0002-9904-1974-13434-8.

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