Heath-Brown–Moroz constant

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


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