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Second Hardy-Littlewood Conjecture

The second Hardy-Littlewood conjecture asserts that for integers x,y>=2,

pi(x+y)-pi(x)<=pi(y),

where pi is the prime counting function (Hardy and Littlewood 1923). In other words, an interval (x,x+y] of length y contains at most as many primes as the initial interval (0,y].

Hensley and Richards (1974) constructed sets of more than pi(y) integers in a half-open interval (a,a+y] whose residues omit at least one class modulo each prime number. The k-tuple conjecture, the first of the Hardy-Littlewood conjectures, predicts that infinitely many translates of such a set consist entirely of primes. Any such translate sufficiently far to the right would violate the second Hardy-Littlewood conjecture.

See also

Hardy-Littlewood Conjectures, k-Tuple Conjecture, Prime Counting Function

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References

Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.Hensley, D. and Richards, I. "Primes in Intervals." Acta Arith. 25, 375-391, 1974. https://doi.org/10.4064/aa-25-4-375-391.Richards, I. "On the Incompatibility of Two Conjectures Concerning Primes; A Discussion of the Use of Computers in Attacking a Theoretical Problem." Bull. Amer. Math. Soc. 80, 419-438, 1974.

Cite this as:

Weisstein, Eric W. "Second Hardy-Littlewood Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SecondHardy-LittlewoodConjecture.html

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