Heath-Brown–Moroz constant

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

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

where p runs over the primes.{{cite journal | last1=Heath-Brown | first1=D. R. | authorlink1=Roger Heath-Brown | last2=Moroz | first2=B. Z. | title=The density of rational points on the cubic surface X₀³=X₁X₂X₃ | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=125 | pages=385–395 | year=1999|doi=10.1017/S0305004198003089 | issue=3| bibcode=1999MPCPS.125..385H | s2cid=59947536 | url=https://ora.ox.ac.uk/objects/uuid:c0ac6ad5-577c-49a7-acdd-72b7f5865cb2 }}Finch, S. R (2003). Mathematical Constants. Cambridge, England: Cambridge University Press.

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

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

References

External links

[http://mathworld.wolfram.com/Heath-Brown-MorozConstant.html Wolfram Mathworld's article]

Category:Mathematical constants

Category:Infinite products