The Heath-Brown–Moroz constant ''C'', named for
Roger Heath-Brown
David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory.
Education
He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervis ...
and
Boris Moroz, is defined as
:
where ''p'' runs over the
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
[Finch, S. R (2003). Mathematical Constants. Cambridge, England: Cambridge University Press.
]
Application
This constant is part of an
asymptotic estimate
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as beco ...
for the distribution of
rational points of bounded
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
on the
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...
''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'' and their
greatest common divisor equal to 1. Then
:
References
External links
Wolfram Mathworld's article
Mathematical constants
Infinite products
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