Q:I have a numerical counter-example. Here are the steps: 1) Beginning with the reverse implication at time = 3:10, where you state that zeta(s) = zeta(1-s^*) <=> zeta(s) - zeta(1-s^*) = 0 <=> sum_1^infinity (1/n^s) - sum_1^infinity (1/n^(1-s^*)) = 0. This is a reverse implication, which means that you need to prove that a implies b and then you need to second prove that b implies a. In this case you have a <=> b <=> c. The reverse implication of a and b is known already. We just need c. Your proof depends on this, and therefore any counter-example that proves that b does not imply c, or the reverse, invalidates anything that follows. Now, you want a numerical value, so use the first non-trivial zero. The numerical value we will use is the first non-trivial zero s = 1/2 + i 14.1347... . 2) check the reverse implications. zeta(s) = zeta(1-s^*) <=> zeta(s) - zeta(1-s^*) = 0 still holds. How about the third? Well, sum_1^infinity (1/n^s) is undefined for the first non-trivial zero. So too is sum_1^infinity (1/n^(1-s^*)). So you have an undefined term subtracted from an undefined term equal to a finite zero, which is a false statement. 3) Analyze: Does zeta(s) - zeta(1-s^*) = 0 imply a false statement? No. Therefore, the implication does not hold, neither does the reverse implication. 4) Was a numerical counter-example used? It was. Was the proof invalidated with a counter-example? It was. I will provide the link where you can send the $10k after you check your math and confirm.
A: Thank you for your comments and feedback. I appreciate your input. The part you mentioned refers to a super-symmetric equation (SSE). It is often assumed that because ∑(n=1 to ∞) 1/n^s = ∞, the expression ∑(n=1 to ∞) 1/n^s - ∑(n=1 to ∞) 1/n^(1-s*) cannot be equal to zero. However, this assumption is false. It is indeed possible for ∞ - ∞ to converge, and one example demonstrating this is the transcendental zeta function, which you can learn more about at https://www.0bq.com/tzf. This function illustrates that the difference of two infinities can result in a finite value in the critical strip. Regarding the super-symmetric equation, there are several proofs available, and you can find two of them referenced in the video (2:58 orange box) or at https://www.0bq.com/se. I acknowledge that further explanation may be required for understanding the SSE. Currently, I am working on a video that will provide include an explanation of this equality. I encourage you to stay tuned for my next video. Also, you can verify the validity of the SSE by considering ∑(n=1 to ∞) 1/n^s - ∑(n=1 to ∞) 1/n^(1-s*) as ∑(n=1 to ∞) (1/n^s - 1/n^(1-s*)), or you can substitute a finite value for 'n' and then take the limit as 'n' approaches infinity. By computing these calculations, you will find that the resulting value is zero where ζ(s)=ζ(1-s*).
Q: In constructing a mathematical proof, one must build upon the axioms that proceed a proposition; that is, the existing theorems of mathematics. The concept is a fundamental principle in mathematics. In mathematics, when an equation involves undefined terms or operations, the equation as a whole is considered to be undefined or meaningless. This principle is based on the foundational rules and definitions of mathematical operations. For example, if an equation involves division by zero, which is undefined, then the entire equation is considered to be undefined. It is a fundamental aspect of mathematical reasoning and the rules that govern the validity of mathematical statements. Now, I have worked (I believe) about 10 minutes on your proof and my rate is $1k / minute. That comes to $10k. I would appreciate you manning up:
A: Let's consider the functions 1/x and csc(x). If someone claims that 1/x - csc(x) is a divergent function at zero, it would not be accurate to declare anyboday as the winner and ignore the fact that 1/x - csc(x) is actually convergent.
The divergence of 1/x and csc(x) individually does not imply that their difference is also divergent. In mathematics, the behavior of the sum or difference of functions can be different from the individual functions themselves. It is essential to analyze the specific properties and behavior of the combined function.
If there is a claim that 1/x - csc(x) is divergent, it needs to be supported by proper analysis and evidence. However, based on the plot , it appears that 1/x - csc(x) is convergent.
Regarding the second part of your statement about the current understanding of mathematics, it is important to note that mathematics is an evolving field. New discoveries and advancements continually shape our understanding of mathematical concepts and principles. If there are any inconsistencies or unresolved issues, mathematicians strive to address and rectify them through further research and development of mathematical theories.
If you have any further questions or if there's anything else I can assist you with, please let me know.
Follow-up video by Jeff Cook on 12/21/23.
Jeff Cook published a video, becuase I had demonstrated how uninformed he is. He calimed that he came up with a numeric counterexample, but I provided a reasonable answer to help him understand. In summary, he claims that the indeterminate form of ∞-∞ is meaningless, where unfortunately I demonstrated how flawed this claim was, leading him to become visibly upset. I observed signs of bigotry and deception when I was talking to him a few month ago. Regarding bigotry, it's not easy; you need to read the entire comments or pay attention to the laughter in this video.
In the above vedio, Jeff initially stated, "I really believed him; he is going to pay," (Time stamp 2:22) and a few seconds later, he said, "I did not believe for a minute.[that I will pay the prize to the winner]" (Timestamp 2:45). So, even he cannot sustain his fabricated and one-sided story for more than a few seconds. I cannot comprehend how a mathematician fails to understand indeterminate form ∞-∞ and supports him. Also, the attendees in Jeff's video call appear to lack mathematical expertise needed for this topic(I mean no offense, just stating my observation). His claim that he is in contact with mathematician in this regard is false. Jeff presents a screenshot from one video and cites comments from another, showcasing a level of sloppiness, inconsistency, and a lack of attention to detail. Yet, Jeff confidently claims to have discovered a counterexample, stating with pride and conviction that many others couldn't find one. I appers that Jeff is fully aware his claim is false, as he makes no effort to explain his neumerical counterexample. He simply states that he has provided it, hoping no one will investigate and take his word. Jeff Cook made several other false claims; however, I would like to respect his right to speak his mind, and I trust the viewer or reader to discern his errors and false claims. After all this, If you believe he is correct, let me know; I will gladly review and offer an additional $10,000 in addition to his reward. And if you are Jeff's friend, please, before getting all worked up, carefully evaluate his claim with a good mathematician to ensure mathematical accuracy.
"I really belive he was going to pay"
"I did not belive for a minute "
My theory regarding this behavior is that I accidentally exposed his foolishness( I mean no offense; I'm using the best words available to convey the idea), and now he holds a grudge against me, as evident in his video and his attempts to distance himself. I formally apologized and clarified that I had no intention to offend. However, after many months, he returned, still laughing, indicating lingering resentment. I have been asked to comment to present my side of the story, so I share this here just in case people wanted to know. Once again, I hope Jeff won't be further upset that I revealed his intentions here, exploiting his short attention span to prove he is a whether deliberate or inadvertent falsehood.( I mean no offense; I'm using the best words available to convey the idea). I hope you understand that Jeff has made several baseless accusations. In return, I believe I have the right to defend myself. I trust that most people working on R.H are intelligent and can discern the truth.If you are a mathematician, I'm almost certain that you can identify Jeff's error. And yes, this is my life. If you aren't a mathematician, please show Jeff's claim to a proficient mathematician and see it for yourself. Again, if you think Jeff's claim is correct, claim your $10K, andJeff will receive his reward. If not, please drop a note in the comments; perhaps you can help him understand.
I have consistently stood against bullying since my chilhood, advocating for both others and myself. I consistently asked in the comments through various methods. I asked Jeff why he thinks this mathematical concept is meaningless. His approach is to engage more people, without presenting any mathematical evidence, and simply insisting on 'Pay up'. I have seen no merit on his claim and It is very sad to see his actions. Also, it's sad that I have clearly invested time to answer Jeff's question, even drawing a graph for him. Yet, Jeff still cannot understand a math cocept (Indeterminate Form) that ∞-∞ can be zero.
Once again, with no intention of causing offense, I am choosing my words to effectively communicate the idea, and I sincerely hope that this note assists Jeff.