The Heath-Brown–Moroz constant C, named for Roger Heath-Brown and Boris Moroz, is defined as
C = ∏ p ( 1 − 1 p ) 7 ( 1 + 7 p + 1 p 2 ) = 0.001317641... {\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.
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 Xi non-negative integers less than or equal to H and their greatest common divisor equal to 1. Then
N ( H ) = C ⋅ H ( log H ) 6 4 × 6 ! + O ( H ( log H ) 5 ) {\displaystyle N(H)=C\cdot {\frac {H(\log H)^{6}}{4\times 6!}}+O(H(\log H)^{5})} .
See also Euler product § Notable constants
References
External links Wolfram Mathworld's article