Distribution of prime numbers.

Nov 2014
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. 
Nov 2011

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.
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Forum Staff
Nov 2006
UTC -5
The result you cite from Erdős-Kalmár 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.
Sep 2014
Portland, Oregon
In both conjectures II and III, you are using <= in the initial statement, but < in your verification statement. While this works, it leaves the reader wondering if one of them was a mistake.

Take a=10^21, b=10^22. sqrt(b)-sqrt(a) >= 1.

Pi(a,b) = 180340017203297174362 according to the values on wikipedia and my subtraction and typing.

Taking Axler (2014) bounds:
Pi(a,b) <= 180340625745474746436 upper(10^22)-lower(10^21)
Pi(a,b) >= 180339408375639669407 lower(10^22)-upper(10^21)

Using Riemann's R function as an approximation I get:
Pi(a,b) ~ 180340017203256473901

Using the very simple approximation n/(log(n)-1) I get:
Pi(a,b) ~ 180264583581749907965

Using the basic Gauss/Legendre estimate n/log(n) I get:
Pi(a,b) ~ 176725893068855722074

Using Conjecture 2 I get:
Pi(a,b) ~ 175759239482714652394

It is not unreasonable, but it isn't computationally useful as the error is larger than even the simplest difference, and well outside the known bounds for this case.
Nov 2014
The symbol ≈ means it is a good approximation need not be asymptotic.
It is true that lack a bit of rigor when expressing my guesses to make it more understandable to mathematicians, when I have a little time, reeditaré the paper with the help of an apostle of rigor.
I should clarify that I am not a mathematician, I'm just an amateur of numbers and I have no formal education.
I live completely out of academia.

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.