First Hardy–Littlewood conjecture explained

First Hardy–Littlewood conjecture
Field:Number theory
Conjectured By:G. H. Hardy
John Edensor Littlewood
Conjecture Date:1923
Open Problem:yes

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 Edensor Littlewood in 1923.[1]

Statement

Let

m1,m2,\ldots,mk

be positive even integers such that the numbers of the sequence

P=(p,p+m1,p+m2,\ldots,p+mk)

do not form a complete residue class with respect to any prime and let

\piP(n)

denote the number of primes

p

less than

n

st.

p+m1,p+m2,\ldots,p+mk

are all prime. Then

\piP(n)\simCP\int

n
2
dt
logk+1t

,

where
k
C
P=2

\prodq

1-w(q;m1,m2,\ldots,mk)
q
\left(1-1\right)k+1
q
is a product over odd primes and

w(q;m1,m2,\ldots,mk)

denotes the number of distinct residues of

m1,m2,\ldots,mk

modulo

q

.

The case

k=1

and

m1=2

is related to the twin prime conjecture. Specifically if

\pi2(n)

denotes the number of twin primes less than n then

\pi2(n)\simC2

n
\int
2
dt
log2t

,

where

C2=2\prodstyle{qprime,\atop

} \left(1 - \frac \right) \approx 1.320323632\ldots

is the twin prime constant.

Skewes' number

See main article: 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

\piP(p)>CP\operatorname{li}P(p),

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

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.

References

Notes and References

  1. Hardy . G. H. . G. H. Hardy . Littlewood . J. E. . John Edensor Littlewood . Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes. . . 44 . 1–70 . 1923 . 44. 10.1007/BF02403921. free . .
  2. Ian . Richards . On the Incompatibility of Two Conjectures Concerning Primes . Bull. Amer. Math. Soc. . 80 . 419–438 . 1974 . 10.1090/S0002-9904-1974-13434-8 . free .