Greetings to all members of this forum.
I've posted a paper in the Arxiv about the distribution of primes.
I sincerely appreciate any comments or suggestions. ï»¿
I think you should number your equations, that would greatly help to make specific comments. I have looked at it very quickly, but one problem I see is that you need to explain what $\approx$ means in many of your equations. This means now your conjecture is not defined uniquely. You need to rewrite either as the difference between two terms that can be bounded, or if this works better for large numbers, then using big O notations or stuffs like this.
The result you cite from ErdÅ‘s-KalmÃ¡r is wrong -- I imagine you miscopied it. the mistake is in the second inequality, though, on which your result does not depend. However, your proof of "the" lower bound (you should have said "a" lower bound) is wrong, because the cited result is not strong enough to prove it. You can derive it from the Prime Number Theorem, though -- it's not wrong, just its proof. But this is a very weak bound, and I don't know why you'd use it when you already have a better one.
Your heuristic in section 2 is reasonable. I don't know how much of an error term to expect, but I expect that it's small enough that the bound in 2.1 holds for large enough numbers. Remember, though, that when intervals are short (polylogarithmic) the heuristic no longer holds -- see Maier 1985 and related literature.
Your work in section 3 is not useful without error terms. Most of the subtractions have results which are subsumed by the error term when you use the PNT. Conjecture II is still reasonable, though, since you require that the difference between a and b is >> sqrt(b) which should suffice. (This is very much conjectural, of course -- we can't even get this with RH.)
The "calculation" sections are very unusual; I assume these are approximations, despite the lack of $\approx$?
Overall the paper looks fine -- non-cranky -- but at a very elementary level.
Thank you for your appreciation.
2.1 must always work, no matter how small the interval.
In Section 3, we may neglect the error term, the goal is not to find the most accurate approximations. Understand that 3 is a track that allows us to understand 3.1.
You are right, the paper is very elementary and easy to understand; but it seems to prove each conjecture exposed is no easy task.
Finally, remember that in my paper I intend to expose a different path to the distribution of primes, which is not involved Riemann Hypothesis
Thanks for taking your time.
The idea is not to find very accurate approximations to pi (a, b), but to find the lower bound for pi (a, b) as shown in 3.1. I ask you to do the calculations for 3.1.
This allows us to answer questions such as what is the maximum difference between a prime number and the nth prime number that happens or precedes ?
Review the special case n = 1 in 5.3.1.