I had discovered that a solution to the problem of adding all the natural whole numbers gives a value of -1/12. That is S = 1 + 2 + 3 + 4 + 5 + ... = -1/12 This is because S = 1 - 1 + 1 - 1 + 1 - ...= 1/2, then an equation can be made that gets the answer of the first set to come out to -1/12. Before you decide to troll this information, maybe you should watch this video; It explains this in more detail, giving the whole proof. I think it is a little outdated. I had heard that this kind of mathematics was taken out of string theory, and string theory no longer uses 26 dimensions as mentioned in some of the related videos. It got me thinking that S = 1 - 1 + 1 - 1 + 1 - ... = 0. That would be because if you started with -1 instead of the +1, you would end up getting -1, 0, -1, 0, -1, 0,... If you took the average of those numbers you would end up getting -1/2 (That would be the same way they got positive 1/2). Then it got me thinking that the range is actually from -1 to +1, so the average of those two numbers would then be 0. I began looking into the Riemann Zeta function. Then it is thought that only values of 1/2 can have a value of zero. Then it made me think that if 1/2 was used instead of 0 in the proof, then you would only end up getting 0 on the complex plain where you get 1/2.
Suppose you think (?) you have solved the RH, how do you get a serious read without being trashed because you don't have a PhD and the "experts" are not going to take you seriously?
The official hypothesis is so garbage, no one will care to comment on it anyways... I guess no one would care to support why adding up all the positive whole numbers would end up giving you a negative fraction equal to -1/12. PhD's are actually looking for a better way to to solve this kind of problems for physics. It is not something that they accept or agree upon to begin with, and a lot of them believe a better method needs to be made to solve this kind of problem. It is not like it has become accepted scientific fact...
Billvon, I am a retired insurance executive, majored in math (go figure), and I have been working on a new calculus, or more accurately, and extension of Lebesgue theory, and I noticed that my theory was a good fit in proving (at least I think I have) the RH. My paper is rather extensive whose results I have used to give a proof of the RH. Actually, I have given three proofs, one using von Mangodt's revision of Riemann's Explicit Formula, one suggested by Edwards, H.M., Riemann Zeta Function, and one proving von Koch's remainder term. Not sure what the next step should be? Suggestions?
If they're complete you should submit them to the appropriate reputable journal. If you're looking for a discussion of your work on the internet then for this site you might ask rpenner to review your proofs. There's a Google Group sci.math https://groups.google.com/forum/m/#!forum/sci.math Maybe somebody there has the scholarship needed to discuss your proofs? I know one other person, besides rpenner, who definitely possesses the scholarship but I'm not sure if he bothers with Internet forums anymore. That would be Mr_Homm. Who is a Professor of math and physics at UW. If you feel comfortable posting a proof at this site I'm pretty sure rpenner would be looking at it.
Good golly. Lets begin by defining \(z_0(n) \equiv \sum_{k=1}^{\infty} k^{-n} \) where n is any integer larger than 1. It follows that \(z_0(n) = \frac{n \int_{0}^{\infty} e^{-y} y^{n-1} dy }{n!} \sum_{k=1}^{\infty} k^{-n} = \frac{1}{(n-1)!} \sum_{k=1}^{\infty} \int_{0}^{\infty} e^{-ku} u^{n-1} du = \frac{1}{(n-1)!} \int_{0}^{\infty} \frac{e^{-u} u^{n-1}}{1 - e^{-u}} du = \frac{1}{(n-1)!} \int_{0}^{\infty} \frac{u^{n-1}}{e^{u} - 1} du \) . Then for any real number larger than 1, we have \(z_1(x) \equiv \frac{1}{\Gamma(x)} \int_{0}^{\infty} \frac{u^{x-1}}{e^{u} - 1} du\) where extends the domain from integers larger than 1 to real numbers larger than 1. Now for every integer larger than 1, we have \(z_1(n) = z_0(n)\) so we have used the tools of analysis to extend a statement about integers to a larger domain. If \(z_0(2) = \sum_{k=1}^{\infty} k^{-2} = \frac{\pi^2}{6} \approx 1.64 \) then \(z_1(\frac{5}{2}) = \sum_{k=1}^{\infty} k^{-\frac{5}{2}} \approx 1.34\) But \(z_1\) is a function of analysis with the nicest possible derivative properties. And it has a unique analytic continuation to the whole complex plane except at 1. That analytic continuation is the Riemann zeta function. \(\zeta(s) = \frac{1}{s -1} \sum_{n=0}^{\infty} \frac{1}{n+1} \sum_{k=0}^{n} (-1)^k \begin{pmatrix} n\\k \end{pmatrix} (k + 1)^{1-s}\) Because \( \sum_{k=0}^{n} \frac{(-1) \begin{pmatrix} n\\k \end{pmatrix} }{k+1} = \frac{1}{n+1}\) this is easy to work with for the case of s=2. \(\zeta(2) = \sum_{n=0}^{\infty} \frac{1}{n+1} \sum_{k=0}^{n} (-1)^k \begin{pmatrix} n\\k \end{pmatrix} (k + 1)^{-1} = \sum_{n=0}^{\infty} \frac{1}{(n+1)^2} = \sum_{k=1}^{\infty} k^{-2} = z_0(2)\). Likewise, when s = -1, we have \(\zeta(-1) = - \frac{1}{2} \sum_{n=0}^{\infty} \frac{1}{n+1} \sum_{k=0}^{n} (-1)^k \begin{pmatrix} n\\k \end{pmatrix} (k + 1)^{2} = - \frac{1}{2} \left( \frac{1}{1}- \frac{3}{2} + \frac{2}{3} \right) = - \frac{1}{12} \) . So while \(\sum_{k=1}^{\infty} k^{-2} = z_0(2) = z_1(2) = \zeta(2)\) is true, \({ \color{red}{ \sum_{k=1}^{\infty} k = z_0(-1) = z_1(-1) }} = \zeta(-1)\) requires a caveat that the sum doesn't converge and the parameter is outside of the domain of \(z_0\) and \(z_1\). But if what is wanted is the analytic continuation -- something that has all the same derivatives as \(z_1\) but a much larger domain, then \(\zeta(-1) = - \frac{1}{12} \) is the answer that is needed.
There are many ways to parse that sum. Two simple ones are: \(S = (1-1)+(1-1)+(1-1) + \dots = 0\) \(S = 1 + (-1 + 1) + (-1 +1 ) + (-1 + 1) +\dots = 1\) By rearranging terms in the sum, we can get lots of other results, too. The fact that the sum depends on the order in which we add terms tells us that the sum does not converge. That is, it has no definite answer. By careful manipulation, it is possible to make it sum to anything you like.
Abel summation for 1 - 1 + 1 - 1 .... goes like this. \( f_0(x) = \sum_{k=0}^{\infty} a_k x^k = \sum_{k=0}^{\infty} (-1)^k x^k = \lim_{n\to\infty} \frac{1 + (-1)^n x^{n+1}}{1 + x} = \frac{1}{1 +x} ; \quad | x | < 1 \\ AbelSum \left( (-1)^k \right) = \lim_{x\to 1^{-}} f_0(x) = \frac{1}{1 + 1} = \frac{1}{2} = \eta(0)\) And for 1 - 2 + 3 - 4 + 5 - 6 + ... goes like this: \( f_1(x) = \sum_{k=0}^{\infty} a_k x^k = \sum_{k=0}^{\infty} (-1)^k (k + 1) x^k = \lim_{n\to\infty} \frac{1 + (-1)^n x^{n+1} (n x+n+x+2) }{(1+x)^2} = \frac{1}{(1 +x)^2} ; \quad | x | < 1 \\ AbelSum \left( (-1)^k (k+1) \right) = \lim_{x\to 1^{-}} f_1(x) = \frac{1}{(1 + 1)^2} = \frac{1}{4} = \eta(-1)\) Abel summation for these two alternating divergent series agrees with value you would get by subtracting a power of two times the Riemann zeta function from itself. This is motivated by a particular trick manipulating the terms of the divergent series as seen in the video. \(\zeta(0) - 2 \zeta(0) = (1 - 2) \zeta(0) = \eta(0) \\ \zeta(-1) - 4 \zeta(-1) = (1 - 4) \zeta(-1)= \eta(-1) \). The Dirichlet eta function, \(\eta(s) = \left( 1 - 2^{1-s} \right) \zeta(s)\), is sometimes called the alternating zeta function.
That is what I read about it, and that is why it sparked my interest. That is the same reason as to what I heard as to why the theory is not accepted. It just looked like the answer to that set should be zero if there was the same amount of one's as there where zero's. If you started with the negative one, you could get the answer zero and negative one. Well if you took the average of all those answers; -1, 0, and 1, you would end up getting zero for your answer. If there was a definite answer and it was zero, then it would mean that the proof for adding all the positive integers is wrong, and it doesn't equal -1/12. Then I don't think that would be such a bad thing as the result is completely nonsensical to begin with. It could mean some of the basis of string theory is wrong, but it has never been proven anyways. If you tried the same proof in the video where S_2 = 0, then you would end up getting: S - S_2 = 4S, and if you replaced S_2 with zero, you would end up getting S - 0 = 4S. Then 0 = 3S, and S = 0.
Why would anyone bother to consider such a ridiculous conjecture? Is it not obvious that a sum of positive integers cannot be negative? Any proof which results in this conclusion is surely a reductio ad absurdum proof that one or more assumptions are invalid.
Dinosaur, please see post #9 where I point out that two steps connect the sum of purely positive numbers, \(z_0(n)\) which is defined only for integers where n > 1 and the Riemann zeta function \(\zeta(s)\) which is defined for the whole complex plane except s=1. Replacing \(z_0(-1)\) (which is not a number) with \(\zeta(-1) = - \frac{1}{12}\) may be justified when the use of \(z_0(-1)\) is some sort of first-order approximation. The devil would be in the details, of which YouTube videos don't have a lot of.
That is the whole reason why I brought it up. If there is actually something wrong with it, then it would need to be fixed. After thinking about it, I think I know why it doesn't work. A set of numbers can be a large range of anything. If you where to manipulate that set according to the numbers of that set, then you could end up getting just about any relation you wanted when you assign that whole set as a variable, like S_1 - S_2 = 4S_1. I think that is why I ended up getting zero for S_1, because zero is a unique number that can make each side of the equation the same even though it has the same variables multiplied by different digits. Then figuring out what sets you end up getting by trying to calculate each number doesn't have a clear relation when assigning that entire set as a variable. It could be that assigning a variable that represents a set is an invalid operation. For instance, I just got an equation where the only number that could fit in such an equation would be zero. Then there is no way you could end up getting zero by adding up all the positive whole numbers either... EDIT: In other words, the relation between the sets representing as variable in an equation would not have the same relation as manipulating those sets by adding all their numbers. The two relations could have nothing to do with each other mathematically. It would be like saying "x" is a dog and "y" is a cat, so you can set up the equation according to how many fleas they have.
It would seem like you would have to write an equation some way that represents a set. Say you had the set S_1 = 1 + 2 + 3 + 4 + 5... Then it would seem like you would need to figure out a way to write an equation that represents that set, like [y] = 0. I will just put "y" in brackets to indicate that it can only be a whole number (something I just made up). Then y can only be a positive number too, so then |[y]| = 0. Then if S_2 was the set S_2 = 1 - 1 + 1 - 1 + 1... It could be represented by the signum function y = sgn(x). https://en.wikipedia.org/wiki/Sign_function Then you would only want to have the whole numbers, so you could say [y] = sgn(x) for simplicities sake. Then to find out what S_1 - S_2 is would require you to subtract the second equation from the first. I would guess you would end up getting something like -sgn(x) = 0, whatever that means... I guess what I am trying to get at here is that if you want to find the relation between two things in different situations then you have to add or subtract the equations. If you just figure something in a set based on the numbers in them and set up an equation from it, then there is no real relation mathematically between the numbers of those sets and the equation generated by assigning those sets as a variable. By making the sets into the form of an equation, it would seem like you could bypass having to assign sets as a single variable, and their relations could possibly mean something.
https://www.wolframalpha.com/input/?i=-sgn(x)=0 Lol, it looks like x=0. I am not that familiar with signum functions so I put it into wolfram. If that way was correct, then it would mean that if you subtracted S_2 = 1 - 1 + 1 - 1 + 1... from S_1 = 1 + 2 + 3 + 4 + 5... then you would just end up getting zero, because that is what you would get from subtracting the lines of equations from each other that would represent them. On second thought, it would seem like the equations would be better represented as [|y = 0|] and [|y = sgn(x)|]. That could more clearly indicate that the equations are sets, since that is the way sets are represented. Then the absolute value would indicate that it is only for the first and second quadrant or the positive "x" values of the equation. Then additional information could be provided about the rules of that set.
