Heath-Brown–Moroz constant

The Heath-Brown–Moroz constant "C", [cite journal|author=D. R. Heath-Brown | authorlink=Roger Heath-Brown | coauthors=B.Z. Moroz | title=The density of rational points on the cubic surface "X"₀³="X"₁"X"₂"X"₃ | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=125 | number=3 | pages=385–395 | date=1999|doi=10.1017/S0305004198003089] named for Roger Heath-Brown and Boris Moroz, is defined as

:$C=prod\_pleft(1-frac\{1\}\{p\}\; ight)^7left(1+frac\{7p+1\}\{p^2\}\; ight),$

which equals 0.00131764115....

This constant is part of an asymptotic estimate for the distribution of rational points of bounded height on the cubic surface "X"₀³="X"₁"X"₂"X"₃. Let "H" be a positive real number and "N"("H") the number of solutions to the equation "X"₀³="X"₁"X"₂"X"₃ with all the "X"_"i" non-negative integers less than or equal to "H". Then

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

References