Heath-Brown–Moroz constant

The Heath-Brown–Moroz constant C,^[1] named for Roger Heath-Brown and Boris Moroz, is defined as

C=\prod _{p}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)=0.001317641...

where p runs over the primes.

This constant is part of an asymptotic estimate for the distribution of rational points of bounded height on the cubic surface X₀³=X₁X₂X₃. Let H be a positive real number and N(H) the number of solutions to the equation X₀³=X₁X₂X₃ with all the X_i non-negative integers less than or equal to H and their greatest common divisor equal to 1. Then

N(H)=C\cdot {\frac {H(\log H)^{6}}{4\times 6!}}+O(H(\log H)^{5}).

References

↑ D. R. Heath-Brown; B.Z. Moroz (1999). "The density of rational points on the cubic surface X₀³=X₁X₂X₃". Mathematical Proceedings of the Cambridge Philosophical Society. 125 (3): 385–395. doi:10.1017/S0305004198003089.

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