Thank you for considering my question. It is a very simple question. But it is one I need to find the answer to for something I am working on.
The question is this:
Is it conceivably, morally, imaginably POSSIBLE for there ever to be more primes from n to 2n than from 0 to n. Like, for example, could there be more primes from 18 to 36 than from 0 to 18 (of course, we know it's not true in this particular case, but I want to generalize it.)
Yes, it seems counter-intuitive to imagine that this would be the case. Yes, it smacks in the face of our immediate reason. But can it be PROVEN? HAS it been proven? And proven by whom?
I couldn't for the life of me try to prove it myself.
It might involve assuming a zero derivative for pi(n), which might be logarithmic. But we know only that it's close to some logarithmic function n/(ln n). Not that it's in fact a logarithmic function itself (or DO we?).
This would be the last step in a very important theorem I am working on. Please keep in mind that I have no degree in anything and have no formal training in mathematics. So any gaping holes in my knowledge should be made understandable to you.