Paradoxes or Contradictions? Exploring the Riemann-Zeta Function and Riemann Hypothesis by Euler’s Identity and Category Theory

The Golden Ratio φ is conventionally taken for its positive value in the algebra form of (1+√5)/2, while its negative value of (1-√5)/2 from the quadratic solution from the definition, (a+b)/a=a/b=φ,a>b>0, has not traditionally drawn much attentions from scientists and researchers, and is usually denoted as the negative inverse of the positive value in the form -1/φ. From the quadratic definition of ϕ, I define 0=(lim┬(n→∞)⁡〖a^(2-n)-b^2/a^n -a^(1-n) b〗 )∩(lim┬(n→∞)⁡〖a^2/b^n 〗-b^(2-n)-a/b^(n-1) )=(lim┬(n→∞)⁡〖1/a^(n-2) -1/(ϕa^(n-2) )+1/(ϕa^n )〗 )∩(lim┬(n→∞)⁡〖ϕ^2/b^(n-2) 〗-1/b^(n-2) -ϕ/b^(n-1) )=∅,a>b>0, whereby the number ϕ is taken multivalued and the Zermelo–Fraenkel set theory is applied. And then with the relation ϕ^2=ϕ+1, I adopt an infinite deduction to summarize the function lim┬(n→∞)⁡〖〖(ϕ^2-ϕ)〗^(1/n) 〗 being either a constant, which I argue against, or a multivalued recurring sequence. The derivative of the series has led to a metric space with the value of e expressed in a complex number form in a unit circle. The results corroborate with the Euler–Mascheroni constant and Gamma Function, and I hypothesize their further extensions to the whole number line with logic completeness.