Distribution of prime numbers.

Jonas Castillo T

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»¿
[URL="http://arxiv.org/ftp/arxiv/papers/1304/1304.5262.pdf"

Dougy

Hi,

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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CRGreathouse

Forum Staff
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.

danaj

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.

Jonas Castillo T

@Dougy.
The symbol â‰ˆ 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, reeditarÃ© 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.

@CRGreathouse.