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Guest wrote:Important Remark: The divergent Harmonic Series, [tex]\sum_{k=1}^{\infty }\frac{1}{k} = \infty[/tex] implies the Fundamental Theorem of Arithmetic, integer [tex]k \ge 1[/tex], [tex]\sum_{k=1}^{\infty }\frac{1}{k^z} = 0[/tex], and the truth of the Riemann Hypothesis.
Important Update:
What is best?
[tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \sum_{i=1}^{ \infty } p_{i } +[/tex] error [tex]= \infty[/tex].
We assume [tex]p_{i }[/tex] is ith positive prime number.
Why?
"We, humans, learn, but Nature knows."
Two Important Reasons:
1. It is logical and consistent that we apply the basic prime number test (refer to the Fundamental Rule/Law of Prime Number Theory above) to each term of the above series excluding the error term since the error term is not a positive integer (right?). The basic prime number test computes [tex]p_{i } ^{ \frac{1}{2} }[/tex] for all [tex]i \ge 1[/tex], and it is used to determine the primality of each appropriate term.
2.
Guest wrote:In a nutshell, [tex]\zeta(z = 1) = \sum_{k=1}^{\infty }\frac{1}{k} = \infty[/tex] if and only if [tex]\zeta(z = \frac{1}{2} ± bi ) = \sum_{k=1}^{\infty }\frac{1}{k^{ \frac{1}{2} ± bi}} = 0.[/tex]
Relevant Reference Link: Prove the Harmonic Series diverges to infinity.
