What are the sup & inf constants for the largest prime gaps?

What are the sup & inf constants for the largest prime gaps?

Postby Guest » Fri Apr 05, 2019 4:39 am

G(X) is the largest gap size between consecutive primes less than or equal to large X.

And for large X we have,

I. [tex]c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X[/tex]

where [tex]0 < c_{1 } < c_{2} < 1[/tex].

Let [tex]C_{1 }[/tex] be the set of values, [tex]c_{1 }[/tex], that satisfy statement I and let [tex]C_{2 }[/tex] be the set of values, [tex]c_{2 }[/tex], that satisfy statement I.

What are sup [tex]C_{1 }[/tex] and inf [tex]C_{2}[/tex]?

Dave.
Guest
 

Re: What are the sup & inf constants for the largest prime g

Postby Guest » Fri Apr 05, 2019 9:56 pm

Guest wrote:G(X) is the largest gap size between consecutive primes less than or equal to large X.

And for large X we have,

I. [tex]c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X[/tex]

where [tex]0 < c_{1 } < c_{2} < 1[/tex].

Let [tex]C_{1 }[/tex] be the set of values, [tex]c_{1 }[/tex], that satisfy statement I and let [tex]C_{2 }[/tex] be the set of values, [tex]c_{2 }[/tex], that satisfy statement I.

What are sup [tex]C_{1 }[/tex] and inf [tex]C_{2}[/tex]?

Dave.


This question is very doubtful since

G(X) [tex]\le log^{2} X[/tex] may be true for large X.

Dave.
Guest
 

Re: Wihat are the sup & inf constants for the largest prime

Postby Guest » Mon Apr 15, 2019 12:15 pm

Guest wrote:
Guest wrote:G(X) is the largest gap size between consecutive primes less than or equal to large X.

And for large X we have,

I. [tex]c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X[/tex]

where [tex]0 < c_{1 } < c_{2} < 1[/tex].

Let [tex]C_{1 }[/tex] be the set of values, [tex]c_{1 }[/tex], that satisfy statement I and let [tex]C_{2 }[/tex] be the set of values, [tex]c_{2 }[/tex], that satisfy statement I.

What are sup [tex]C_{1 }[/tex] and inf [tex]C_{2}[/tex]?

Dave.


This question is very doubtful since

G(X) [tex]\le log^{2} X[/tex] may be true for large X.

Dave.


Our work to answer this question is worthwhile and ongoing.

Dave.
Guest
 

Re: What are the sup & inf constants for the largest prime g

Postby Guest » Tue Apr 16, 2019 4:30 pm

FYI: On the Convergence of the Largest Prime Gaps, G(X), [tex]c_{1 } *[/tex]G(X), and [tex]c_{2} *[/tex]G(X):

We recall the following statements and question:

G(X) is the largest gap size between consecutive primes less than or equal to large X.

And for large X we have,

I. [tex]c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X[/tex]

where [tex]0 < c_{1 } < c_{2} < 1[/tex].

Let [tex]C_{1 }[/tex] be the set of values, [tex]c_{1 }[/tex], that satisfy statement I and let [tex]C_{2 }[/tex] be the set of values, [tex]c_{2 }[/tex], that satisfy statement I.

What are sup [tex]C_{1 }[/tex] and inf [tex]C_{2}[/tex]?
_____________________________________________________________________________________________________________

Plot of G(X) = [tex]log^{2} X[/tex], over closed interval, [100, 30000],

https://develop.open.wolframcloud.com/app/objects/28edc87e-7575-436d-a0b0-10ae36b39bcf#sidebar=compute

Assuming [tex]G(X) \rightarrow log^{2}[/tex]X as X [tex]\rightarrow \infty[/tex].

We expect sup [tex]C_{1 } \rightarrow[/tex] inf [tex]C_{2 } \rightarrow[/tex] 1 as X [tex]\rightarrow \infty[/tex].

Dave.
Guest
 

Re: What are the sup & inf constants for the largest prime g

Postby Guest » Wed May 01, 2019 4:30 pm

Are you confused about the true meaning of inf and sup?
Guest
 

Re: What are the sup & inf constants for the largest prime g

Postby Guest » Wed May 01, 2019 4:57 pm

Guest wrote:Are you confused about the true meaning of inf and sup?

Oops! If the prior statements are wrong, please switch sup and inf appropriately, or accept a new definition of inf (least/inferior upper bound) and sup (greatest or superior lower bound). Either way, have it your way. Okay?
Guest
 

Re: What are the sup & inf constants for the largest prime g

Postby Guest » Sat May 04, 2019 12:57 pm

Guest wrote:FYI: On the Convergence of the Largest Prime Gaps, G(X), [tex]c_{1 } *[/tex]G(X), and [tex]c_{2} *[/tex]G(X):

We recall the following statements and question:

G(X) is the largest gap size between consecutive primes less than or equal to large X.

And for large X we have,

I. [tex]c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X[/tex]

where [tex]0 < c_{1 } < c_{2} < 1[/tex].

Let [tex]C_{1 }[/tex] be the set of values, [tex]c_{1 }[/tex], that satisfy statement I and let [tex]C_{2 }[/tex] be the set of values, [tex]c_{2 }[/tex], that satisfy statement I.

What are sup [tex]C_{1 }[/tex] and inf [tex]C_{2}[/tex]?
_____________________________________________________________________________________________________________

Plot of G(X) = [tex]log^{2} X[/tex], over closed interval, [100, 30000],

https://develop.open.wolframcloud.com/app/objects/28edc87e-7575-436d-a0b0-10ae36b39bcf#sidebar=compute

Assuming [tex]G(X) \rightarrow log^{2}[/tex]X as X [tex]\rightarrow \infty[/tex].

We expect sup [tex]C_{1 } \rightarrow[/tex] inf [tex]C_{2 } \rightarrow[/tex] 1 as X [tex]\rightarrow \infty[/tex].

Dave.


Note: Please accept our new definition of inf (least or inferior upper bound) and sup (greatest or superior lower bound).
Guest
 


Return to Number Theory



Who is online

Users browsing this forum: No registered users and 4 guests