Luboš Motl
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 Mar 6 comment Why does $1+2+3+\cdots = -\frac{1}{12}$? Sure, it's a completely analogous sum. Just like $1+2+3+\dots$ quantifies the ground state energy in 1+1 dimensions (of a string), $1+1+1+\dots$ quantifies the ground state's charge (of a system of free fermions, for example). It's really the reason why the two degenerate states $|0\rangle$ and $c_0|0\rangle$ have ghost numbers $\pm 1/2$, for example. Feb 10 comment Why does $1+2+3+\cdots = -\frac{1}{12}$? It might be but it is a wrong conclusion. The key property of this $\infty$, which may be subtracted by a counterterm, is that it is independent of all independent variables, so it is formally a scale-invariant constant, and $0$ is the number to be associated with it. The fact that the sum of integers equals $-1/12$ is perhaps the most important fact about mathematics among the laymen - and among mathematicians without breadth and depth. Feb 8 answered Equivalence of Two Lorentz Groups Jan 24 comment $1 + 1 + 1 +\cdots = -\frac{1}{2}$ Dear @Pedro, I am saying that $\zeta(2k+1)$ are transcendental numbers because I am ready to bet all my wealth on the validity of the claim. After all, even $\pi^n$ are transcendental numbers for rational nonzero values of $n$ so even zeta of even positive integers are (provably) transcendental. But the zetas of odd positive integers are (I claim without a rigorous proof) much more transcendental than the powers of $\pi$. I never claimed that I had a rigorous proof of the transcendentality but if you want to claim that I said something incorrect, you should have a proof which you don't have. Jan 24 comment $1 + 1 + 1 +\cdots = -\frac{1}{2}$ Nope, it's just believed by me and almost everyone that they're transcendental. The much "easier" proof that zeta(3) is irrational came just recently, in 1978, see en.wikipedia.org/wiki/Ap%C3%A9ry's_theorem Jan 23 comment $1 + 1 + 1 +\cdots = -\frac{1}{2}$ Nice, +1. An alternative calculation of $\zeta(0)=-1/2$ appears early in this article: motls.blogspot.com/2014/01/… - See my comment under the OP's question for 4 more articles about the topic if you wish... Jan 23 comment $1 + 1 + 1 +\cdots = -\frac{1}{2}$ Five more articles with 8 different ways to compute the sum of integers (most of the methods extend to the "sum of ones", too) etc.: motls.blogspot.com/2007/09/… motls.blogspot.com/2011/07/… motls.blogspot.com/2014/01/… motls.blogspot.com/2014/01/… motls.blogspot.com/2014/01/… Sep 20 comment The inverse Fourier transform of $1$ is Dirac's Delta Well, I think you would also have to discuss that the negative regions of $(\sin x)/x$ don't combine to other distributions like $\delta'$ etc. But otherwise OK, I upvote it. Jun 25 comment Fourier transform of a signal sequence? I would be sort of surprised if there is an analytic formula for the Fourier transform of this contrived function. Jun 18 awarded Good Answer May 29 revised How find the number of $z$,such that$|a^2-b^2-b+1|\le 10$ added 365 characters in body May 29 answered How find the number of $z$,such that$|a^2-b^2-b+1|\le 10$ May 28 answered Evaluate integral in terms of Gamma function May 24 revised Grover Algorithm Orthogonal vectors added 15 characters in body May 24 comment Grover Algorithm Orthogonal vectors Sorry, I made a mistake. I meant that $\omega$ and $s'$ are orthogonal. My answer is hopefully fixed. I don't understand the question "how I will be able to define $\sqrt{N}|\omega\rangle$". It's just multiplication, right? One should know how to multiply vectors by numbers. May 24 answered Grover Algorithm Orthogonal vectors May 7 awarded Yearling Feb 3 awarded Enlightened Feb 3 awarded Nice Answer Dec 15 answered Fourier transform vs Fourier series