Heath-Brown–Moroz constant

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

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

where p runs over the primes.^[1][2]

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

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

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