Conjecture on Upper Bound of # of Ratios Under a Given Tp Height

Pick an arbitrary positive rational q. Let's say that S is the set of all m, where m=(2*floor[Tp(q)/log2(n)]+1) for all primes n<=2^Tinf(q).

Lets also say that C is the set of all rationals x such that Tp(q)>=Tp(x), where infinity>=p>=1.

I conjecture that |C| is less than or equal to the product of (1-1/(2*p^j) and all elements in the set S (if S is empty, let's assume the product is 1), where j is a constant between 0 and 1. The strong version of this conjecture is when j=1.

I've found ways to prove this for a few specific cases, but I can't seem to prove it is true in general. Can anyone provide insight on how to go about proving this for all cases?

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> Pick an arbitrary positive rational q. Let's say that S is the set of all m, where m=(2*floor[Tp(q)/log2(n)]+1) for all primes n<=2^Tinf(q).

I think I wrote this part wrong. That last "Tinf(q)" should probably Tp(q).

> I conjecture that |C| is less than or equal to the product of (1-1/(2*p^j) and all elements in the set S (if S is empty, let's assume the product is 1), where j is a constant between 0 and 1. The strong version of this conjecture is when j=1.

Lets only consider the strong version where j=1, although I have suspicions that j could be a unique constant much lower than 1.

Ryan