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a New Math Tool

There are great indications that direct or general solutions for ∞=∞ will have a significant impact on mathematics, like the invention of “algebra,” “calculus,” and “e.” 

 Below are simple definitions that, most of the time, we can use  ∞-∞ form to determine ∞=∞ form.  

The point of the above rules are as follows:

1. This indeterminate form is valid for function only.

2. Functions can diverge into three types of infinity (∞, z∞, ∞~). The absolute value of these infinities is a real number that increases endlessly. 

3. We can call two infinities equal if the difference of those infinities converges to zero.

4.  We can call two infinities not equal if the difference of those infinities converges to a non-zero value. They are not equal if the difference diverges to ∞ or z∞ or ∞~.

5. If the difference of infinities alternate between the above two forms, it will be an indeterminable or undecided form. For examples: ∞~ - z∞ or ∞ - ∞~.    


The Riemann Hypothesis and a New Math Tool (a new Indeterminate form)

We know that 1+1+1…=∞=-1/2=𝜁(0) and 1+2+3…=∞=-1/12= 𝜁(-1) , and since 1+1+1…≠ 1+2+3…, 𝜁(0) ≠ 𝜁(-1), -1/2≠-1/12 in this case ∞≠∞. This shows that ∞ is not always equal ∞.

∞-∞ and ∞/∞ expressions are indeterminate forms, and due to the nondeterministic nature of ∞=∞ , it makes sense to consider it a new category of indeterminate form.

Considering the transcendental zeta function, you can see ∞=∞ is satisfied iff 𝜁(s)=0.

Also, we can use the ABC zeta function to prove that the for zetazeros ∞-∞=0 and ∞=∞ are equivalent.

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