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

[Guest]

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]

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]

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]

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]

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

[Guest]

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]

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).