sevensixtwo Posted September 30, 2018 Author Posted September 30, 2018 Image from these undergrad complex analysis notes (page 5).
taeto Posted September 30, 2018 Posted September 30, 2018 (edited) So in the article, the real numbers are divided into those x with |x| at most R and those with |x| > R. Does the numerical value of the radius R matter at all? It is not specified in the OP article? What is the meaning of \( \widehat{\infty} \)? Edited September 30, 2018 by taeto
sevensixtwo Posted September 30, 2018 Author Posted September 30, 2018 1 hour ago, taeto said: What is the meaning of ∞ˆ ? That's a good question. I was trying to find the answer in the OP image, and it looks like there's a sentence and a citation for that very question. You should have a look and tell me if it answers your question. 1 hour ago, taeto said: Does the numerical value of the radius R matter at all?∞ˆ I think the numerical value of the radius of the neighborhood of the origin might not exist due to equation (1). What is your opinion regarding the question you asked?
Strange Posted September 30, 2018 Posted September 30, 2018 41 minutes ago, sevensixtwo said: I was trying to find the answer in the OP image, and it looks like there's a sentence and a citation for that very question. You should have a look and tell me if it answers your question. Are you saying that this is not your paper? (Just curious, I assumed it was, and that was the reason you posted it.) The paper is here: http://vixra.org/abs/1809.0557 (also not sure why you posted an image, which is hard to read and quote from) The references are: [1] http://vixra.org/abs/1809.0234 [2] http://vixra.org/abs/1703.0073
taeto Posted September 30, 2018 Posted September 30, 2018 (edited) Thank you Strange for the references! So we know only a little bit about \(\widehat{\infty}\) from that paper, in which it is a central quantity. Then for the OP: How can we learn enough about it to be able to read past the third line of the paper, where it first appears? Could you present an explicit example in which \(\infty\) and \(\widehat{\infty}\) both appear in the roles that they are supposed to occupy in this paper? 1 hour ago, sevensixtwo said: I think the numerical value of the radius of the neighborhood of the origin might not exist due to equation (1). What is your opinion regarding the question you asked? The value of R goes into the definition of "neighborhood" in the beginning. Are you saying that this value does not exist, hence the definition is meaningless? If you think that the paper begins with meaningless definitions, then why would you present it here in the forum? Edited September 30, 2018 by taeto
sevensixtwo Posted September 30, 2018 Author Posted September 30, 2018 (edited) 40 minutes ago, taeto said: Are you saying that this value does not exist, hence the definition is meaningless? No. The meaning is that ascribed to it in the sentence which describes it in the body of the article and the author, who is definitely me, has also provided a citation for anyone who cannot parse "all canonically non-standard properties except additive absorption." However, anyone who cannot parse that statement is probably not able to parse the similar statements in the citation. On the other hand, if you wanted to make the argument that something which has not been previously invented cannot be invented then I would be open to reading your argument. Edited September 30, 2018 by sevensixtwo
wtf Posted September 30, 2018 Posted September 30, 2018 (edited) 1 hour ago, sevensixtwo said: No. The meaning is that ascribed to it in the sentence which describes it in the body of the article and the author, who is definitely me, has also provided a citation for anyone who cannot parse "all canonically non-standard properties except additive absorption." However, anyone who cannot parse that statement is probably not able to parse the similar statements in the citation. On the other hand, if you wanted to make the argument that something which has not been previously invented cannot be invented then I would be open to reading your argument. I couldn't parse it. Can you please explain it? What do you mean by "canonically non-standard properties?" Are you referring perhaps to the nonstandard real numbers aka the hyperreals? If so, you should definitely say so. And even with that interpretation I have no idea which canonically non-standard properties you're referring to. Edited September 30, 2018 by wtf
Strange Posted September 30, 2018 Posted September 30, 2018 I am not a mathematician and am not qualified to comment on the content of the paper, but I dod have a (perhaps naive) question. The paper says: Quote there are as many real numbers in the neighborhood of infinity as there are non-zero real num- bers in the neighborhood of the origin As the reals are uncountable, can one make a statement comparing the numbers in one subset with those in another like this? Just curious...
wtf Posted October 1, 2018 Posted October 1, 2018 (edited) 1 hour ago, Strange said: I am not a mathematician and am not qualified to comment on the content of the paper, but I dod have a (perhaps naive) question. The paper says: As the reals are uncountable, can one make a statement comparing the numbers in one subset with those in another like this? Just curious... It's certainly true in the usual interpretation of "a" neighborhood of infinity; if by that we mean the set of reals whose absolute value is greater than some fixed value. In other words there's a bijection between the set of reals x with |x| <= 1000 and the set of reals x with |x| > 1000. But the OP seems to have some other definition in mind for "the" neighborhood of infinity, the definition of which he has not given with sufficient clarity. Edited October 1, 2018 by wtf
sevensixtwo Posted October 1, 2018 Author Posted October 1, 2018 1 hour ago, wtf said: I couldn't parse it. You are probably not in the intended audience for the paper then. -2
wtf Posted October 1, 2018 Posted October 1, 2018 Just now, sevensixtwo said: You are probably not in the intended audience for the paper then. It's true. I'm not a member of the empty set. Why don't you just explain yourself? 2
taeto Posted October 1, 2018 Posted October 1, 2018 9 hours ago, sevensixtwo said: No. The meaning is that ascribed to it in the sentence which describes it in the body of the article and the author, who is definitely me, has also provided a citation for anyone who cannot parse "all canonically non-standard properties except additive absorption." However, anyone who cannot parse that statement is probably not able to parse the similar statements in the citation. On the other hand, if you wanted to make the argument that something which has not been previously invented cannot be invented then I would be open to reading your argument. I would argue that you cannot refer to "the neighbourhood" of infinity, when your definition of such a neighbourhood clearly depends on a positive real number R, unless you fixate the value of R. Grammatically correct would be "a neighbourhood" otherwise. But since every nonzero complex number belongs to some such neighbourhood, it would seem to add no further information. 1
swansont Posted October 1, 2018 Posted October 1, 2018 9 hours ago, sevensixtwo said: You are probably not in the intended audience for the paper then. ! Moderator Note But you posted it here for discussion, so they are the intended audience of the post. Which makes that response less than acceptable.
studiot Posted October 1, 2018 Posted October 1, 2018 I would like to see the OP expand upon the definition of the words neighberhood and infinity as used in this discussion. Does infinity refer to the point at infinity appended to the complex plane or what? for instance different infinites are distinguished in analysis http://www.math.harvard.edu/~nate/teaching/UPenn/2009/spring/math_360/lectures/week_8/lecture_15/lecture_15.pdf Similarly if we go into complex analysis and Riemanain geometry then the definition of neighberhood becomes more difficult https://books.google.co.uk/books?id=vmZ6PVtaexwC&pg=PA64&lpg=PA64&dq=neighborhood+of+the+point+at+infinity&source=bl&ots=q0RdNEtJvz&sig=5U8PU-dLmiY226_VN0_P8CmWDOA&hl=en&sa=X&ved=2ahUKEwiRyaHJjOXdAhVlAsAKHc6ICgkQ6AEwBHoECAUQAQ#v=onepage&q=neighborhood of the point at infinity&f=false
studiot Posted October 1, 2018 Posted October 1, 2018 (edited) Then further down we have the statement that there are as many real numbers in the neighborhood of infinity as there are non zero real numbers in the neighborhood of the origin, (ie in the deleted neighborhood of the origin) Surely it is true that any neighborhood in the reals has the same cardinality as any other since they are dense. So what's the big deal? Finally he comes at the end to state that if he can find a number, off the critical line, whose zeta function is zero then he has disproved the theorem. That is self evident. But he has to show that (at least) one such number exists. Edited October 1, 2018 by studiot
sevensixtwo Posted October 2, 2018 Author Posted October 2, 2018 (edited) On 9/30/2018 at 7:08 PM, Strange said: As the reals are uncountable... I say there is the same number of them because each uniquely labelled R_0 number has a hat R number with that same label, and there are no other hat R numbers except for those which share a label with R_0 numbers. In the paper I call this label "b" 14 hours ago, studiot said: Does infinity refer to the point at infinity appended to the complex plane or what? If you would have read the paper, you would have seen in the first few lines that infinity is "+/- infinity" and if you were familiar with analysis at even the undergraduate level you would know that these are the endpoints of the extended real line. You might feel more at home in the less advanced forum because these symbols I use are already quite basic. Edited October 2, 2018 by sevensixtwo -5
taeto Posted October 2, 2018 Posted October 2, 2018 14 hours ago, sevensixtwo said: If you would have read the paper, you would have seen in the first few lines that infinity is "+/- infinity" and if you were familiar with analysis at even the undergraduate level you would know that these are the endpoints of the extended real line. You might feel more at home in the less advanced forum because these symbols I use are already quite basic. I believe the point is that it is already known that every nontrivial zero has real part strictly between 0 and 1 (the 'critical strip'). So where you say that there are nontrivial zeros with real part equal to \(-\widehat{\infty}+b\), we all see that as a consequence, \(b\) must lie strictly between \(\widehat{\infty}\) and \(\widehat{\infty}+1\). How do we make sure that \(b\) is not exactly equal to \(\widehat{\infty}+1/2\), in which case you would just have a point on the critical line, and not a counterexample?
sevensixtwo Posted October 3, 2018 Author Posted October 3, 2018 If you would have read the paper, you would have seen that "b" is a real number. There is no real number greater than infinity. Please read the whole paper. There are only a few sentences and I put all of them in there because they are important. Also, I have proven that there are non trivial zeros outside of the critical strip in the neighborhood of infinity. If you believe there is a proof that there are no such zeros then please post it. That would be really great for me to study because such a result would have a large impact on my research. -1
taeto Posted October 4, 2018 Posted October 4, 2018 9 hours ago, sevensixtwo said: If you would have read the paper, you would have seen that "b" is a real number. There is no real number greater than infinity. Isn't \( b - \widehat{\infty}\) the real part of your \(z_0\)? If \(b\) is also real then \(\widehat{\infty}\) is the difference of two real numbers, hence real. It is what you call "infinity"? But for any real number there are larger real numbers. And anyway, the question is why the real part \( b - \widehat{\infty}\) of your \(z_0\) is not \(1/2\). Is it because it is not real?
swansont Posted October 4, 2018 Posted October 4, 2018 10 hours ago, sevensixtwo said: If you would have read the paper ! Moderator Note That would be the paper you didn't provide a link to? And ignores our rule that people need to be able to participate in the discussion without clicking on any links? No, that does not wash. If clarification is needed, you must provide it here.
studiot Posted October 4, 2018 Posted October 4, 2018 On 02/10/2018 at 2:26 AM, sevensixtwo said: If you would have read the paper, you would have seen in the first few lines that infinity is "+/- infinity" and if you were familiar with analysis at even the undergraduate level you would know that these are the endpoints of the extended real line. You might feel more at home in the less advanced forum because these symbols I use are already quite basic. Well I did read it but I am having trouble right where you point to at the beginning with the assertion made in equation 1, as shown in my attachment. This is why I asked for clarification.
sevensixtwo Posted October 4, 2018 Author Posted October 4, 2018 (edited) 11 hours ago, taeto said: Isn't b−∞ˆ the real part of your z0 ? yes 11 hours ago, taeto said: If b is also real then ∞ˆ is the difference of two real numbers, hence real. That doesn't follow. If you think it does, please give two real numbers whose difference is infinity. Before you do that, please consider that if you define a real number as numeral rather than as a cut in the real number line I will have to remind you that the definition of a real number is that it is a cut in the real number line. 10 hours ago, swansont said: ! Moderator Note That would be the paper you didn't provide a link to? That would be the paper I posted in full in OP post of this thread. Here is link, if you would like to click it. 10 hours ago, studiot said: This is why I asked for clarification. I'm not sure what I could clarify for you without simply copying a sentence from the paper. Is everything clarified for you now? I'm not able to discern any request for clarification in your most recent post. Maybe you are asking if "b" can be a real number in the neighborhood of infinity. The answer is no; I have used "R" in the usual way to refer only to numbers in the neighborhood of the origin without "hat 0" inserted for clarity. I have not proposed to modify what we use the symbol "R" for, I have only proposed to modify the concept of real numbers. In the future people might consider "R" to be the union of "R_0" and "hat R" but until they all agree to change what "R" means I will use it in the regular way: "R" = "R_0" without the "hat 0" symbol. Edited October 4, 2018 by sevensixtwo
studiot Posted October 4, 2018 Posted October 4, 2018 (edited) 1 hour ago, sevensixtwo said: Is everything clarified for you now? Why on earth would you ask this? Why do you think I posted my last post if everything was clarified ? Specifically what is [math]\hat 0[/math] ? I ask because the very first sentence in your proof is ambiguous, and I posted my attempt to make sense of it, upto and and including equation 1. Therefore I had to choose one of the possible interpretations of your first sentence. So do you agree with my posted reasoning? 1 hour ago, sevensixtwo said: Maybe you are asking if "b" can be a real number in the neighborhood of infinity. The answer is no Your answer will be interesting since my post is a proof of this statement. Edited October 4, 2018 by studiot
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