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

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

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

And for large X we have,

I. $$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

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

What are sup $$C_{1 }$$ and inf $$C_{2}$$?

Dave.
Guest

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

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. $$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

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

What are sup $$C_{1 }$$ and inf $$C_{2}$$?

Dave.

This question is very doubtful since

G(X) $$\le log^{2} X$$ may be true for large X.

Dave.
Guest

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

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. $$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

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

What are sup $$C_{1 }$$ and inf $$C_{2}$$?

Dave.

This question is very doubtful since

G(X) $$\le log^{2} X$$ 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

FYI: On the Convergence of the Largest Prime Gaps, G(X), $$c_{1 } *$$G(X), and $$c_{2} *$$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. $$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

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

What are sup $$C_{1 }$$ and inf $$C_{2}$$?
_____________________________________________________________________________________________________________

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

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

Assuming $$G(X) \rightarrow log^{2}$$X as X $$\rightarrow \infty$$.

We expect sup $$C_{1 } \rightarrow$$ inf $$C_{2 } \rightarrow$$ 1 as X $$\rightarrow \infty$$.

Dave.
Guest

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

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

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

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

Guest wrote:FYI: On the Convergence of the Largest Prime Gaps, G(X), $$c_{1 } *$$G(X), and $$c_{2} *$$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. $$c_{1 } * log^{2} X \le G(X) \le c_{2 } * log^{2} X$$

where $$0 < c_{1 } < c_{2} < 1$$.

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

What are sup $$C_{1 }$$ and inf $$C_{2}$$?
_____________________________________________________________________________________________________________

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

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

Assuming $$G(X) \rightarrow log^{2}$$X as X $$\rightarrow \infty$$.

We expect sup $$C_{1 } \rightarrow$$ inf $$C_{2 } \rightarrow$$ 1 as X $$\rightarrow \infty$$.

Dave.

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