Heath-Brown–Moroz constant

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

[math]\displaystyle{ C=\prod_p\left(1-\frac{1}{p}\right)^7\left(1+\frac{7p+1}{p^2}\right) = 0.001317641... }[/math]

where p runs over the primes.

Application

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

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

External links

• Wolfram Mathworld's article