Guest wrote:Keywords: Prime Number Theorem (PNT)
Hmm. What are the number of primes less than or equal to X?
What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?
Thus, for the largest prime gap...,\beta G(X), we have simply (via the great PNT),
[tex]G(X) \le \beta(X) * log X[/tex] where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where [tex]2 << \beta(X) << X[/tex].
Guest wrote:An Update:
Keywords: Prime Number Theorem (PNT)
Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?
What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?
Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),
[tex]G(X) \le \beta(X) * log X[/tex]
where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where [tex]log X \le \beta(X) << \pi(X)[/tex].
Guest wrote:An Update:
Keywords: Prime Number Theorem (PNT)
Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?
What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?
Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),
[tex]G(X) \le \beta(X) * log X[/tex]
where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where
[tex]log X \le \beta(X) << \pi(X)/log X[/tex].
Guest wrote:Guest wrote:An Update:
Keywords: Prime Number Theorem (PNT)
Hmm. What is the number of primes less than or equal to X which we denote as [tex]\pi(X)[/tex]?
What is the average gap size, [tex]g(X)_{average }[/tex], between all consecutive primes less than or equal to X?
Thus, for the largest prime gap..., G(X), we have simply (via the great PNT),
[tex]G(X) \le \beta(X) * log X[/tex]
where [tex]\beta(X)[/tex] tends to infinity as G(X) tends to infinity and where
[tex]log X \le \beta(X) << \pi(X)/log X[/tex].
Hmm. For large X,
is G(X) more or less [tex](log X)^{2}[/tex]?
Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
Power Series Expansion of [tex](log X)^{2}[/tex]:
For large X, we have
[tex](log X)^{2} = (log X)^{2} -2(logX)\sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}})^{2}[/tex].
If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?
Or if [tex]G(X) < (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?
What is the function, k(X)? This is a difficult and deep question!
Guest wrote:Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
Power Series Expansion of [tex](log X)^{2}[/tex]:
For large X, we have
[tex](log X)^{2} = (log X)^{2} -2(logX)\sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}} + ( \sum_{k=1}^{\infty}\frac{(-1)^{k}}{kX^{k}})^{2}[/tex].
If [tex]G(X) \rightarrow (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?
Or if [tex]G(X) < (log X)^{2}[/tex] as [tex]X \rightarrow\infty[/tex], how are k and X related?
What is the function, k(X)? This is a difficult and deep question!
Hmm. The problem is deep but not difficult.
FYI: We can approximate log X with the appropriate Harmonic Series...
Relevant Reference Link:
[url]https://en.m.wikipedia.org/wiki/Euler–Mascheroni_constant[/url]
Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
Guest wrote:Hmm. That's theory! We want to see an example.
Let nth prime, [tex]p_{n }[/tex] = 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.
According to Wikipedia on prime gaps,
https://en.m.wikipedia.org/wiki/Prime_gap#Simple_observations,
we have,
[tex]p_{n+1 } - p_{n }[/tex] = 8350.
Since [tex]X = p_{n } \approx p_{n+1}[/tex],
we compute,
[tex]\pi(X)/log^{2}(X) \approx[/tex] .00002523 * [tex]\pi(X)[/tex].
http://m.wolframalpha.com/input/?i=PrimePi%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227%29%2F%28log%28293703234068022590158723766104419463425709075574811762098588798217895728858676728143227.%29%29%5E2
Wow! Theory works very well!
Guest wrote:Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
We can do better! For large X we have,
[tex]c_{1 }(log X)^{2} \le G(X) \le c_{2 }(log X)^{2}[/tex]
where [tex]0 < c_{1 } < c_{2} \le 1[/tex].
Moreover, [tex]\pi(X)[/tex]= O[tex](log^{2}(X))[/tex].
Guest wrote:Guest wrote:Guest wrote:For large X,
[tex]log X << G(X) < (log X)^{2}[/tex].
This is our best result!
We can do better! For large X we have,
[tex]c_{1 }(log X)^{2} \le G(X) \le c_{2 }(log X)^{2}[/tex]
where [tex]0 < c_{1 } < c_{2} \le 1[/tex].
Moreover, (1) [tex]\pi(X)[/tex]= O[tex](log^{2}(X))[/tex].
Note:. We changed o-notation to O-notation.
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