Efficient Goldbach Partitions Formula with Intuitive Reasoning

[Marchal]

Intuitive reasoning has led me to develop a simple approximation, which contains factors different from those used in well knoen formulas. Numerically, "my" formula delivers results, which are almost as accurate, as Hardy-Littlewood`s with the Shah-Wilson correction.

Thanks in advance for any comments

 

[CRGreathouse]

The formula (2) has a typo or is otherwise ill-posed. I trust from the parenthetical comment that the intent is

$$\frac12\pi(n)\prod_{3<p<\sqrt n}\frac{p-2}{p-1}$$

Your approximations in (2.1) are off by a factor of about 2.245838. (I'll let you figure out where this came from.) This causes the derivation of (2) to be wrong.

Asymptotically, the ratio of g_alt/g_HL is not 1 but about 1.123. The first factor is 1 + O(1/log n), the third factor is 1 + o(1), and looks to drop off like 1 + O(1/log^2 n). The second factor is where the trouble comes from.

 

[Marchal]

Thank you indeed, CRGreathouse, for signalling this error.

Using Merten's Theorem, I find, that g_alt must be divided by 2*exp(-gamma)=1,122918967...,
with gamma=0,577215664901... being Euler-Mascheroni's constant.

Then, g_alt will be asymptotically equivalent to g_HL

I'm now working on correcting the derivation of (2)

Best Regards - Marchal

 

[Marchal]

Here is a revised version of my article.

 

[CRGreathouse]

I wrote several paragraphs of response but the forum ate it, grr. In short: There are about two dozen errors in the first few pages, but they're mostly minor and correctable. By page 3 you use notations that are not only nonstandard but for which I can't even find any valid interpretation.

Unfortunately your use of ≈ rather than, say, ~ makes your statements non-testable (and even non-falsifiable in a Poperian sense).

 

[Marchal]

Dear CBGreathouse

Thanks for reviewing my text with so much attention!.

In standard notation I should write

n$$\neq$$0 (modp) (equation 2a)

q$$\neq$$0 (modp) and q$$\neq$$n (modp) (equation 2.2.1)

I changed the equivalence symbols, as you suggested (see attachment).

 

[Marchal]

Did I miss uploading the file Goldbach3?

 

[CRGreathouse]

1.2 still has issues. It's C_2 not C_HL and the "<= infinity" should be dropped. (There are rare cases where you want to write "< infinity" for clarity, but this isn't one and certainly "<= infinity" is just wrong.)

2a confuses me; what's the definition of g_alt? If this is supposed to be the definition you need = not ~ (or one of the defined-as symbols, if you prefer).

For congruences you need ≡ ≢ not = ≠ .

Have to go now; might look at p. 3 ff. later.

 

[Marchal]

I still worked on page 3. Here it comes with corrections.

 

[Marchal]

Please ignore my last message. Here comes page3-improved version.

 

[CRGreathouse]

2.2.2 is the first questionable part: you claim that one obtains asymptotic equivalence. (You mark it an estimate, but then write ~.) This heuristic has been well-known for hundreds of years, but a proof is lacking. This is what you'd expect if the primes fell 'randomly', but it's not clear that they do in an appropriate fashion.

 

[Marchal]

Thanks, Greg. I fully agree. Besides, I suspect my deduction to be substantially erronous. I'll need some time to clarify the matter.
Until then
Marchal

 

[Marchal]

Here I am again! For correct formula & deduction, see APPENDIX attached.