Riemann's Last Theorem
vs.
The Riemann Hypothesis
The primary justification for referring to Riemann Hypothesis(RH) as the Riemann's Last Theorem(RSLT) lies in our contributions to its verification. Riemann's Last theorem posits that once a hypothesis garners proof, it attains the status of a theorem. Now that this theorem has been conclusively proven, it is entirely fitting to designate it as the 'last theorem' in Riemann's legacy, and this nomenclature serves as a mark of respect—hence, it is aptly named Riemann's Last Theorem.
We acknowledge that it's a widely accepted colloquialism to say "Riemann hypotheses Proof or Riemann hypotheses Solved," which conveys the essence of our understanding. However, to put it plainly, neither mathematically nor grammatically is it correct to phrase it as such. Within the realm of rigorous mathematical discourse, referring to it in that manner would be incorrect and imprecise.
Another elegant reason is that we can consider a differences between RH (Riemann Hypothesis) and RSLT (Riemann's Last Theorem), which can make the explanation more concise . In the figure below, you can observe the distinction in Riemann's context. Specifically, it depicts the transition from 'ζ(s) ≠ 0' to 'Re(s) ≠ 1/2' and vice versa(Re(s) ≠ 1/2 => ζ(s) ≠ 0 to ζ(s) ≠ 0 => Re(s) ≠ 1/2 ), as relevant within the given context. This transition can be linked to the curious result of '1+2+3…=-1/12,' where we delve into more detailed considerations than the first option we encounter. This beautiful transition, step by step, transforms something expected into an unexpected result, akin to '1+2+3...=-1/12.' This is just one of the many techniques that have been employed in the proof of the Riemann Hypothesis. Although the information here is less detailed compared to Riemann's Last Theorem article, it should provide sufficient clarity.
The RSLT statement says that if the real part s is “NOT” 1/2, then the zeta function “CANNOT” be zero. Also holds that if the real part of s is 1/2, then the zeta function CAN be zero. ( negating “NOT” and “CANNOT”) .
Important Note: Riemann's Hypothesis is proven through Riemann's Last Theorem, which employs a clever method for analyzing and establishing connections between two sets: the set of zeros of the zeta function (B) and the set of values of the zeta function on the critical line (A). This theorem demonstrates that the set of zeros of the zeta function is a subset of the values of the zeta function on the critical line. In simpler terms, Riemann's Last Theorem introduces additional elements to the previously irregular and unpredictable set of zeros of the zeta function, transforming it into a regular and predictable set. Subsequently, it proves a specific property for the values on the critical line. Riemann's Last Theorem shows that this specified condition applies to any subset of this new set of values.
RSLT( Riemann's Last Theorem) Vs RH (The Riemann Hypothesis) notes on the M2.2 proof:
To prove the Riemann Hypothesis, it is unnecessary to require ζ(s)≠0 => Re(s)≠½ where ζ(s) ≠ ζ(1-s*) . However, there are compelling reasons to go the extra mile, some of which are listed here.
In the context of Riemann's last theorem article, if ζ(s)≠0 then the RH holds true, no further explanation or poof is needed to justify the R.H. simply because the goal is to study the zeroes of zeta function. In other words, we define variable “s” belongs to the set of none trivial zeroes of the zeta function. Re(s) ≠ 0.5 => ζ(s) ≠ 0 implies the validity of the Riemann Hypothesis ( https://youtu.be/VAAZrYXWb_0?si=wMFiaS9pFy3JL_eB&t=229).
It's crucial to note that Riemann's Last Theorem specifically addresses the zeros of the zeta function within the critical strip. We needn't worry about trivial zeros or whether ζ(s) ≠ 0.
So why we opt to use apparent harder path to prove RH by using ζ(s) ≠ 0 while our focus is actual on ζ(s) = 0.
The reason is it's impossible to prove RH using ζ(s) = 0. Consider the picture below. We have to choose a harder path from the beginning because we need a path to prove RH. This is like an investment to prove RH. It's like inventing a very expensive spaceship to go to Mars rather than buying a very fast luxury airplane that can get us off the ground much cheaper and faster with zero cahnce to get us to the Mars. The point is, ζ(s) = 0 => Re(s) = ½ has shown no promise that we can prove RH. On the other hand, the six steps below generate redundant proofs that yield very compelling results.
1. ζ(s)= 0 => Re(s) = ½ (RH)
2. Re(s) ≠ ½ => ζ(s) ≠ 0 (RSLT)
3. Re(1-s*) ≠ ½ => ζ(1-s*) ≠ 0 (RSLT)
4. ζ(s) ≠ ζ(1-s*) (RSLT)
5. ζ(s) & ζ(1-s*) ≠0 (RSLT)
6. ζ(s) ≠ 0 => Re(s) ≠ ½ (RSLT)
Note that 4 is correct because of its prior step 2, where Re(s) ≠ ½ (1=>2=>3=>4=>5=>6). It might seem redundant. However, there is one major benefit: that we use in 5. we don’t know the exact locations of the zeros of the zeta function; we only know they exist somewhere on the critical line and are highly unpredictable. On the other hand, ζ(s) ≠ ζ(1-s*) holds is related to the entire critical line, which is very predictable. 5 gives us this statement: The set of zeroes of the zeta function is a subset of set of s where 4 is satisfied. We think it needs to be said because, once we have proven that condition 4 holds for Re(s) ≠ ½ , then it implies that the zeta function cannot be zero because either both ζ(s) & ζ(1-s*) are zero or neither of them is. Steps 1 to 6 proofs is akin to viewing a cube from different angles; for many, it might seem redundant, while for others, seeing all six sides is necessary.
Important note this area of math where we obtain results like 1 + 4 + 9... = 0. There are hidden logical steps involved in obtaining zero, and it only makes sense in context.
Steps above 6, and in particular, steps 4 and 5, together provide the correct meaning for 6. Without considering them, the 6 doesn't make sense, much like the seemingly contradictory result of 1 + 2 + 3... = -1/12.
Regarding some common point that some equation like 1 + 4 + 9 + ... =0 is not true. That is the traditional view. It is just an illusion, akin to seeing the Earth as flat. In the correct perspective, those values coincide, meaning 1 + 4 + 9 + ... = ∞ coexists with 1 + 4 + 9 + ... =0. If you say one is wrong, you are implying that both are wrong and if you believe one is correct, the other one follows, just like 2 and 6 above. We haven’t published any proof beyond the critical strip because not needed complexity, but we have published a simple video and presented the proof in the RSLT article. For example, we have shown that 1^.5 + 1/2^.5 + 1/3^.5 + … = ∞ follows 1^.5 + 1/2^.5 + 1/3^.5 + = -1.46…. In other words, they are the same thing ( Step Zero of Analytic Continuation Gateway to the Riemann Hypothesis ). However, like the flat Earth and Geocentric ideas, the truth may appear very counterintuitive to us.
Please observe the mesmerizing transformation bewween A and B sets in the captivating illustrations below.
Let be the set of complex numbers s such that Z={s ∈ ℂ | 0<(Re(s) <1}
Let A ⊂ Z be the set of complex numbers s such that Re(s) = 1/2: A = {s ∈ Z | Re(s) = 1/2}
Let B ⊂ Z be the set of complex numbers s such that ζ(s) = 0: B = {s ∈ Z| ζ(s) = 0}
~A =Z-A , ~B=Z-B , B ⊂ A
We can use the sets Z-A = ~A and Z-B = ~B to establish a two-way connection for the zeros of the Riemann zeta function, which is the primary focus of the Riemann Hypothesis. To clarify, we can carefully utilize ~A and ~B to establish connections among the equations related to Riemann's Last Theorem (RSLT). Please note that this is not a proof of these equations; it's a map that explains the connection between RSLT and the Riemann Hypothesis.
Riemann Hypothesis (R.H.) : For all non-trivial zeros ζ of the Riemann zeta function ζ(s), if ζ(s) = 0, then Re(s) = 1/2. (s ∈ B)
Invers of Riemann Hypothesis(~R.H.) means: For all non-trivial zeros ζ of the Riemann zeta function ζ(s), if ζ(s) ≠ 0, then Re(s) ≠ 1/2.(s ∈ {~ A ∩ ~B})
Notice that nagetion of s ∈ {~ A ∩ ~B}=~A in this context is B sinc we can say { A ∩ B} =B.( this true if RSLT ture and for the proof you need to see Riemann's Last Theorem Article. To put it simply, we aim to demonstrate that when the real part of 's' is not equal to 1/2 (Re(s) ≠ 1/2), then the Riemann zeta function, ζ(s), is also not equal to 0. We don't need to work on if ζ(s) = 0 on the critical line (Re(s) = 1/2), becuase it will fulfill R.H. objective. Essentially, we need to prove that there are no non-trivial zeros of ζ(s) off the critical line( this is it) . This is crucial in establishing the Riemann Hypothesis (RH).
Please observe that the red or blue paths individually prove the Riemann Hypothesis (please see Riemann's Last Theorem articel).
Converse:
- Original Statement: P → Q
- Converse: Q → P (Negation: If Q, then P)
Inverse:
- Original Statement: P → Q
- Inverse: ¬P → ¬Q (Negation: If not P, then not Q)Contrapositive:
Contrapositive:
- Original Statement: P → Q
- Contrapositive: ¬Q → ¬P (Negation: If not Q, then not P)
In these representations:
- "¬" represents negation.
- "→" represents implication or "if...then..."
- "P" and "Q" are placeholders for propositions or statements.
It's important to note that the contrapositive is always logically equivalent to the original statement, meaning they have the same truth value. The converse and inverse may or may not have the same truth value as the original statement.
Note 1 : We can limit/or extend the set of values to ensure the truth of the inverse and converse. For example, if you are currently in Los Angeles, it means you are in California, which implies you are in the United States, and ultimately on planet Earth within the solar system. However, since all people live on Earth, we can confidently state that if you are in the solar system, you are indeed living on Earth. This statement holds true based on the assumption that we know it's a fact that all people reside on, above, or within Earth .In this context, people correspond to non trivial zeros of the zeta function. Earth represents the critical line, and the solar system symbolizes the critical strip.
Note 2 :We need to consider the entire set including any hidden elements, when applying logic. It's crucial to encompass the full scope of elements under consideration before applying logic to accurately establish logical relationships.
Question: Wouldn't 0.5 be a straight up counterexample as Zeta(0.5) ≠ 0, but Real(0.5) = 0.5, thus the equivalence you're claiming is wrong.
Answer: The short answer is no because ζ (0.5) = ζ (1−0.5) and, and it's necessary that ζ(0.5) ≠ζ(1−0.5). Also, note that Re(s)≠0.5 => ζ(s)≠0, is enough to prove the Riemann Hypothesis https://youtu.be/VAAZrYXWb_0?si=wMFiaS9pFy3JL_eB&t=229.