Constant of motion: Difference between revisions
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The '''Heath-Brown–Moroz''' constant ''C'',<ref>{{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''<sub>0</sub><sup>3</sup>=''X''<sub>1</sub>''X''<sub>2</sub>''X''<sub>3</sub> | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=125 | pages=385–395 | year=1999|doi=10.1017/S0305004198003089 | issue=3}}</ref> named for [[Roger Heath-Brown]] and [[Boris Moroz]], is defined as | |||
:<math>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 [[Prime number|primes]]. | |||
This constant is part of an [[asymptotic estimate]] for the distribution of [[rational point]]s of bounded [[height function|height]] on the [[cubic surface]] ''X''<sub>0</sub><sup>3</sup>=''X''<sub>1</sub>''X''<sub>2</sub>''X''<sub>3</sub>. Let ''H'' be a positive real number and ''N''(''H'') the number of solutions to the equation ''X''<sub>0</sub><sup>3</sup>=''X''<sub>1</sub>''X''<sub>2</sub>''X''<sub>3</sub> with all the ''X''<sub>''i''</sub> non-negative integers less than or equal to ''H'' and their [[greatest common divisor]] equal to 1. Then | |||
:<math>N(H)= C \cdot \frac{H(\log H)^6} {4\times 6!} + O(H(\log H)^5).</math> | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Heath-Brown-Moroz constant}} | |||
[[Category:Mathematical constants]] | |||
{{numtheory-stub}} | |||
{{num-stub}} | |||
Revision as of 19:48, 22 October 2013
Template:Primary sources The Heath-Brown–Moroz constant C,[1] named for Roger Heath-Brown and Boris Moroz, is defined as
where p runs over the primes.
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
References
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