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aNewMathTool

**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.**