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5:01 PM
'The following words are attributed to Hilbert:
“If I were to awaken after sleeping for a thousand years, my first question would be:
Has the Riemann Hypothesis been proven?”'
 
Anonymous
@bolbteppa Whoa, I vaguely remember someone saying that but couldn't remember who.
 
Anonymous
Hilbert, nice
 
Anonymous
BTW my knowledge of RH is limited to Wiki. But I found that video really enlightening ^
 
Think I tried to watch it and gave up, will try again
 
5:04 PM
@Blue I think it's decent so far. I think it's meant for people with a bit of QM knowledge and little SR and tends to go on the safe side, so it can be slow at times. I got a similarly titled book on LQG and it was in the complete opposite direction. Like a couple chapters went from "this is a Lorentz transformation" to "this is how we define a Yang-Mills theory"
 
Anonymous
@danielunderwood Hah, so it doesn't need/presume GR?
 
The Riemann zeta function is heavily connected with primes
(there's this one key connection between complex analysis and number theory: Residue theorem)
it's used to prove the prime number theorem, that there are approximately $\frac x {\ln x}$ primes below $x$
 
I've got to do a presentation about stars next week, anyone know about protostars?
 
@JohnRennie not sure what it was, but something jogged this from my memory: theguardian.com/politics/2014/nov/27/…. Wasn't 2014 fun?
 
I gotta make sense of them
 
5:10 PM
We define the prime counting function $\pi(x) := \#\{p \le x \mid p~\text{prime}\}$, and the prime number theorem states $\pi(x) \sim \frac x {\ln x}$
to prove that, we introduce some auxiliary functions
$\psi(x) := \displaystyle \sum_{p \le x} \log p \left\lfloor \frac {\log x}{\log p} \right\rfloor$ and $\psi_1(x) := \displaystyle \int_0^x \psi(t) \ \mathrm dt$
because these functions are easier to work with than the $\pi$ function
 
@Blue Not that I've seen. It does use tensor index notation that people with some knowledge of GR may be more comfortable with, but I haven't seen anything other than that. I don't think the book actually treats gravity much other than the weak field limit $g = \eta + h$. I got a preview of Polchinski from amazon and it looks like it requires GR and QFT, which is a bit out there for me at the moment
 
and it turns out that $\psi_1(x) = \displaystyle -\frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{x^{s+1}}{s(s+1)} \frac {\zeta'(s)} {\zeta(s)} \ \mathrm ds$
and this is one of the examples of the connections between Riemann zeta function and prime numbers
@Blue ^
 
Anonymous
@LeakyNun Very interesting
 
Anonymous
I need to read it a few more times
 
Anonymous
But getting the idea
 
5:16 PM
@EmilioPisanty "ingrained liberal bias"...come to the US and you can party like it's 2014 again
 
Anonymous
How exactly do they get that form for $\Psi_1(x)$ tho?
 
@danielunderwood it's just how much easier it was to take things in good fun given how much lower the stakes were
I mean, all sorts of things went wrong in 2014, but they mainly happened in other places
 
Why can't number theory be easy and obvious, even simple things are impossible :(
 
Crimea & whatnot
it's much easier to go "look at those guys, if you gave them any power they'd break democracy" than it is to make fun of those same guys after they got that power and the went ahead and broke democracy
(* caveats apply. obviously.)
 
Anonymous
@danielunderwood I see. Could give it a try then :P
 
5:20 PM
but anyways. let's not talk politics. or at least I'll stop here.
 
@Blue A generalization of "functions like $\zeta$" is called Dirichlet functions, i.e. functions that are given by a series in the form of $\sum f(n) n^{-s}$ (for $\zeta$ we have $f(n) = 1$). And it turns out that you can differentiate a Dirichlet function term by term, and multiplying two Dirichlet functions together can be performed using Dirichlet convolution (e.g. $(f \ast g)(10) = f(1) g(10) + f(2) g(5) + f(5) g(2) + f(10) g(1)$). Using that knowledge, we convert $\frac{\zeta'}{\zeta}$ into
a Dirichlet series, and then interchange integral with summation
and we express $\psi_1$ also in terms of a series
and then use some Residue theorem to prove that they're equal
 
@Blue yeah I've picked up a handful of books that were like "woah, how am I going to get through this?" and it certainly wasn't one of those
 
Anonymous
@LeakyNun Interesting. So residue theorem is the key connection here
 
20 mins ago, by Leaky Nun
(there's this one key connection between complex analysis and number theory: Residue theorem)
Anyway, from here:
13 mins ago, by Leaky Nun
and it turns out that $\psi_1(x) = \displaystyle -\frac1{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{x^{s+1}}{s(s+1)} \frac {\zeta'(s)} {\zeta(s)} \ \mathrm ds$
you can use some heavy estimation to bring that integral to a small loop around $s=1$
The $\dfrac{x^{s+1}}{s(s+1)}$ gives $\dfrac{x^2}{2}$ (since this is non-zero finite number, we can move it out)
 
Anonymous
Ah, I see
 
5:29 PM
the residue of $\dfrac{\zeta'(s)}{\zeta(s)}$ at $s=1$ is $-1$, by the argument principle, since $\zeta$ has a simple pole at $s=1$
so the whole integral becomes $\dfrac{x^2}2$
 
Anonymous
I should probably go through the individual steps though. Know of any place where this is explained in full?
 
so $\psi_1(x) \sim \dfrac{x^2}2$
and with less work we can establish the prime number theorem from this
 
Anonymous
Makes sense
 
@Blue there are quite a lot of proofs out there, and I've looked through many, and I prefer chapter 2 of this paper
I don't know about your taste, but you might like this "short proof"
I don't prefer short proofs because it gives me not much understanding
 
Anonymous
Awesome, thanks :) I don't know much number theory, but since this is more of complex analysis, it seems I can give it a try.
 
Anonymous
5:33 PM
(Will ping you if I get stuck)
 
@Blue glad to help anytime
be warned that the whole proof is estimation upon estimation upon estimations
 
Anonymous
Hehe
 
Anonymous
Will keep that in mind
 
6:00 PM
@Blue This table ^ from here might give you more of an understanding of the connection between prime numbers and Riemann zeta function
$\sigma$ is the real part of the input of the Riemann zeta function
(the convention is $s = \sigma + it$)
(you won't believe how long I tried to re-find this table)
 
6:15 PM
Iirc the reflection formula for the zeta function guarantees that if $\sigma + it$ is a zero, then so is $1-\sigma+it$ is as well. (I think you need t nonzero for this to work, so you don’t get doubling of the trivial zeros on the negative real line)
 
Does anyone know what happened to this guy? physics.stackexchange.com/users/12951/mark-rovetta He stopped to login to all stack exchange websites on the same october 2015 date
his profile indicates he wanted to be active forever
 
So there being no zeros for sigma>1/2 is enough to ensure that all nontrivial zeros also have 1-sigma<1/2, and therefore sigma = 1/2
Which is the more familiar form
 
@vzn Huh?
 
@coniferous_smellerULPBG-W8ZgjR there's a facebook profile that looks like the same person, with updates from 2017
so it doesn't look like a catastrophe
 
thanks @EmilioPisanty
 
6:20 PM
ditto LinkedIn
@coniferous_smellerULPBG-W8ZgjR have you given any thought to removing the MathJax compilation error from your Location field?
 
yes, but unfortunately I don't really like the idea to change my location. 1st because it is harmless, the only downside is 1 cm of lost space in the page where my name is listed, which is nowhere near a catastrophe. 2nd: if that's really troublesome, I hope this makes dev actually fix the bug (assuming it is a bug). if I simply comply, they might just not fix it and who knows what other people can do with that bug, assuming it is one
but if that gets me banned, sure I'll comply
 
@Semiclassical that's right
 
and btw it was just a random string of chars, as I once wrote to blue in this chat @EmilioPisanty , you can search for "random" or "zsh" I suppose if you want the details
I had 0 intention to "break" anything
 
@coniferous_smellerULPBG-W8ZgjR it's not a bug. You can't disable MathJax because other users do have intentional MathJax in their location profiles.
the location field's length breaking the formatting, on the other hand, is indeed a bug, and it's on the devs to fix it.
seriously, though - if you're going to include active characters in the charset you're randomizing from, wouldn't it make more sense to escape those active characters?
The only change that's required is changing the $s to \$ and they will display as 'intended'.
 
0
Q: May I ask a big-list question?

QuasilatticeWhile writing a meta-section in my thesis I wanted to give some examples of important theory proposals preceding experimental verification but I couldn't name very many from the top of my head. For example, I knew that Wolfgang Pauli proposed the existence of the neutrino which was experimentall...

 
6:31 PM
just as a suggestion.
 
vzn
6:46 PM
@Avantgarde just replying to your own question, do you (really) care or not?
7 hours ago, by Avantgarde
What is the meaning of life?
 
35
Q: Sir Michael Atiyah's conference on the Riemann Hypothesis

José Hdz. Stgo.An announcement of a conference by Sir Michael Atiyah that is to take place next Monday between 9:45 and 10:30 is circulating in several websites. Sure it should always be a thing to listen to him, but this announcement stands out in that he is claiming to have a simple proof of the Riemann Hypot...

 
user280247
Are all facts which allow us to postulate electrons as waves, related to many electrons?
 
user280247
Also in what regards to photons, i.e beam of light
 
user280247
Cause it seems to me that an electron is a particle, not a wave, but many electrons can behave as waves. Is this correct?
 
user280247
(btw, I know abt de broglie, etc etc)
 
6:52 PM
Motl says in disq.us/p/1vxfaj1 he e-mailed Atiyah and gives the response he got
 
@santimirandarp No. The double slit experiment, for instance, also works when only one electron at a time passes through the slits.
 
Anonymous
@LeakyNun Aha, cool!
 
user280247
@ACuriousMind where did you read it? Some days ago I saw a proof on the contrary, I can show you it
 
user280247
 
Anonymous
@santimirandarp Show it?
 
user280247
6:56 PM
show the proof? @Blue
 
Anonymous
Yes
 
user280247
Yes, thats on wikipedia
 
user280247
XD
 
Anonymous
Where?
 
Anonymous
Link?
 
user280247
6:57 PM
@Blue look at the link I appended
 
@santimirandarp That's it. You can see that even though only one electron at a time passes, the interference pattern still builds up.
So it is indeed the wavefunction of each individual electron interfering with itself.
 
user280247
No, but each dot is an electron @ACuriousMind
 
Anonymous
@santimirandarp I can only see an image, which confirms that even a single electron can create inteference
 
@santimirandarp Yes, each dot is an electron. What's your point?
 
user280247
@Blue I interprete it as the contrary
 
user280247
6:58 PM
@ACuriousMind the interference pattern arises with many particles
 
user280247
@ACuriousMind so just one electron behaves as a particle, many as a wave...
 
Anonymous
@santimirandarp Read the section on "Interference of individual particles"
 
user280247
@Blue but am I wrong?
 
Anonymous
> Sending particles through a double-slit apparatus one at a time results in single particles appearing on the screen, as expected. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (see the adjacent image).
 
Anonymous
@santimirandarp Yes
 
Anonymous
7:00 PM
If individual electrons behaved as particles there wouldn't be any interference pattern
 
@santimirandarp No - each electron behaves as a quantum object, which is neither a classical particle nor a wave. What the single-electron double slit shows is that you see the interference pattern even if no electron can interact with any other electron. You may need many measurements of the single electrons to see the patterns, but these did not interact, so each of them individually must have a wavefunction that can interfere with itself.
 
user280247
hmmm I see...I supposed this experiment was made differently
 
Anonymous
Try throwing paint balls at a wall and check if you get any interference pattern
 
Anonymous
(or use slits if you want to)
 
user280247
@Blue well don't be rude...it's just a question
 
7:02 PM
Perhaps a more drastic demonstration is that the interference pattern vanishes when you add "which-way information" to each electron by additionally measuring directly at the slit.
 
Anonymous
@santimirandarp Umm...what?
 
Anonymous
I meant it for real
 
He apparently made 2 claims that were questioned recently, not just the sphere one...
 
user280247
@ACuriousMind that's surprising (for me at least)
 
Anonymous
A delayed-choice quantum eraser experiment, first performed by Yoon-Ho Kim, R. Yu, S. P. Kulik, Y. H. Shih and Marlan O. Scully, and reported in early 1999, is an elaboration on the quantum eraser experiment that incorporates concepts considered in Wheeler's delayed-choice experiment. The experiment was designed to investigate peculiar consequences of the well-known double-slit experiment in quantum mechanics, as well as the consequences of quantum entanglement. The delayed-choice quantum eraser experiment investigates a paradox. If a photon manifests itself as though it had come by a single path...
 
Anonymous
7:06 PM
Actually read the second one
 
Anonymous
It's basically an extension of the YDSE
 
user280247
So single electrons produce an interference pattern. If there is only one photon we will see just one dot isn't it?
 
Anonymous
If you allow them to build up, yes
 
Anonymous
The point is that they can interfere with themselves
 
user280247
Yes, or more plainly they behave as waves @Blue
 
Anonymous
7:08 PM
@santimirandarp Well, you're missing the point
 
Anonymous
Yes, if you send a single electron you'll notice one dot on the screen
 
Anonymous
BUT
 
user280247
with many, an interference pattern, I know
 
Anonymous
If you send several electrons you'll notice an interference pattern building up
 
Anonymous
NOW
 
Anonymous
7:09 PM
the point is, if a single electron behaved as a particle
 
Anonymous
this wouldn't be possible
 
user280247
Yes, and neither do photons, isn't it?
 
@santimirandarp The thing is that quantum objects are neither particles nor waves. But when you make a position measurement of one, like when it hits a detector screen, you force it to look like a classical particle that has a definite position.
 
Anonymous
@santimirandarp Yeah, neither electrons nor photons behave as particles (in this context)
 
user280247
Hmm I don't like that @ACuriousMind
 
7:10 PM
@santimirandarp Thankfully, nature has never cared for what we humans like ;)
 
user280247
@ACuriousMind XD
 
user280247
Just want to say it
 
Anonymous
Measurement is a bit confusing to understand though (intuitively)
 
user280247
All I've been taught is: look it is an interference pattern, so they behave as waves
 
Anonymous
That's non-sequitur really
 
user280247
7:12 PM
do you mean it is an excuse?
 
user280247
I have just one more question...
 
Anonymous
I mean the fact that they behave as waves doesn't follow directly from the fact that they form interference patterns on the screen :P
 
Anonymous
Anyhow, that's good for a first physics class
 
user280247
No, that's exactly what was confusing...
 
user280247
yes, and it was physics for chemists so it is not very deep
 
user280247
7:15 PM
So the first electron is happily going towards the slit
 
user280247
then it interferes and we measure
 
user280247
most probable way to find the dot would be the center of the screen isn't it?
 
user280247
I hope to be clear
 
user280247
And the chemist was fired off
 
Anonymous
7:23 PM
10
A: What is the wavefunction of the Young Double Slit experiment?

user12029Well, there are many things you could do. You could: consider two gaussian beams (the linked article is for electrodynamics) apply some paraxial approximation (which would be more appropriate to treat electrons with a high forward momentum) do a cheap/cheesy symmetric point source approximation...

 
user280247
@Blue :)
 
user280247
not related at all but this video is really cool: youtube.com/…
 
user280247
from edward teller, about schrodinger's cat
 
Anonymous
@santimirandarp That's actually a good question
 
Anonymous
12
A: Double Slit experiment with just one photon or electron

Alan Rominger Has anyone done an experiment where they send just one particle? (And end the experiment) Do you observe two points on the screen in that case or just one? You don't observe two points - you get a probability. This is what the video was trying to indicate by showing dots on the screen. Tho...

 
user280247
7:29 PM
thanks, I can't read it right now but I've favorited them
 
Anonymous
YDSE is something I get a new perspective for, every time I revisit it
 
Anonymous
So, yeah, thanks for the questions!
 
Anonymous
@santimirandarp One thing to emphasize I guess is: There isn't anything special about two slits. You could use 200 slits and still you'd get a single dot on the screen when you send a single electron
 
7:46 PM
@EmilioPisanty are you here?
I haev figured out a way to bug the userpage, it's like 2 km long now. BUT
I have noticed that there is an CSS limit nevertheless. it's just set way too high
I mean, look at the userpage. completely demolished physics.stackexchange.com/…
 
Anonymous
@coniferous_smellerULPBG-W8ZgjR Nice discovery XD
 
I haven't discovered it, it's @EmilioPisanty who did
 
@vzn So the meaning of life is given in that novel?
 
hopefully this will make the dev lower that CSS limit
 
I know the novel. It's pretty popular, though I haven't read it myself.
 
Anonymous
7:50 PM
@Avantgarde Dude it's even given in my pocket dictionary
 
Anonymous
(except that it is too fat to actually fit in my pocket)
 
Man it's pretty cool how matlab has transfer functions and feedback connections built-in
 
@coniferous_smellerULPBG-W8ZgjR that.... is one heck of a bug.
@coniferous_smellerULPBG-W8ZgjR Keep in mind that it's a weekend. This is unlikely to get dev-team attention before Monday.
it might be a good idea to try and catch them on the MSE chat then, though.
 
8:12 PM
just want to say thank you @EmilioPisanty . I'll leave it as it is... for now.
 
no worries.
 
MSE is math stack exchange? or main?
 
@coniferous_smellerULPBG-W8ZgjR the main meta
 
8:28 PM
Oh, Atiyah proved the Riemann hypothesis?
I'm late to the show.
 
Why is it that when drawing a root locus, the value on $K$ for which the systems starts oscillating is always in the interval on the real axis (pertaining to the root locus) closest to origin?
 
@Avantgarde claims to
 
@EmilioPisanty Yeah I read he'll present it on Monday.
 
he's scheduled to present the proof at the Heidelberg Laureate forum, the talk's on Monday morning
whether he has a proof or an almost-a-proof-but-not-quite, remains to be seen
I hope he has a proof
 
Like here
 
8:31 PM
@Lozansky roots for what?
 
Is it obvious (a priori) $K$ is in the interval $[-1,0]$ and not in $[-3,-2]$?
@EmilioPisanty Poles to the closed system's transfer function
 
@Lozansky no idea what you're talking about.
 
Yeah sounds interesting, though I don't know anything about the hypothesis
 
what closed system, and what transfer function?
 
@EmilioPisanty The transfer function is $G_c(s) = \dfrac{K(s+2)}{s(s+1)(s+3)+K(s+2)}$
 
8:33 PM
@Lozansky if you're hoping for replies from folks that don't already have their noses soaked in exactly the same formalism as you are, that's nowhere near enough information to start helping you.
 
Anonymous
@Lozansky Well, the image shows the oscillation of what? What does the transfer function represent/correspond to physically?
 
@EmilioPisanty I'm bad at explaining, but most of the formalism can be found here: en.wikipedia.org/wiki/Root_locus (look at definition)
So basically we find poles to the transfer function as a function of $K$
 
"1. Let's assume that GR time dilation is caused by the laws of thermodynamics. " "2.Now let's try the opposite, let's say that GR time dilation is caused by the laws of thermodynamics. " LOL
the opposite? Looks the same to me....
 
@Blue Oscillations to a system that looks like this:
And $G_o(s) = \dfrac{K(s+2)}{s(s+1)(s+3)}$
The root locus is a tool to figure out what values of $K$ make this system stable. What I don't understand is how to find $K$ such that the system becomes oscillating
 
oh f*s
please stop
 
8:40 PM
@EmilioPisanty I like the pun
 
@Lozansky what pun?
 
"In light of the advance[...]" :P
 
@Lozansky nah, that's just bait
flattery
 
But it's clever bait. A hook with a pun on it
 
I don't see the pun, sorry
 
8:44 PM
What journal are they from?
 
@Lozansky there are two roots on [-1,0] and a single root on [-3,-2]. The oscillations start when the two roots meet. A single root won't shoot off into nonzero imaginary part.
@Lozansky they're pushing "American Journal of Optics and Photonics" this time
I've had multiple emails from SciencePG
they go straight in the bin so I probably can't tell whether the previous ones were pushing the same journal
the ones from the past three months are. before that, the records are gone.
 
@EmilioPisanty Hmm makes sense I guess. Do you understand how they found $K=0.4186$?
And the pun was that a journal on optics writes "In light of the advance" :P
 
@Lozansky that's when the two roots meet.
(assuming you're plotting roots. if you're plotting poles, take the reciprocal.)
when roots meet, you get a double zero
double zeros have a zero derivative
so if you're plotting the roots of some $f_K(s)=0$, the value $K=0.4186$ corresponds to the point where $f_K(s)=0$ and $f_K'(s) =0$ have a joint solution.
 
9:02 PM
@EmilioPisanty Yup, that does it
 
@Lozansky ;-)
 
9:20 PM
hmmmmmmm
@DavidZ @dmckee Do you know if there is a record on Physics Meta of when (and why) [books] questions were banned?
i.e. the change described here
 
10:08 PM
@EmilioPisanty I actually thought they were still on topic albeit somewhat "controlled" - they just go by the name of now. But perhaps dmckee could clarify what he meant there. I might be missing something.
 
Anonymous
5
Q: New book policy: What should we do about the old questions?

ManishearthWe have a new book/resource recommendation policy. However, it is of a form that it can only be applied to new questions. We have many old questions with answers that do not satisfy the policy, what should we do about them?

 
Anonymous
I suppose dmckee was referring to this
 
Anonymous
Note: For now, if I come across old book questions (example), I'm closing them with the intent of revisiting them once this discussion gets over. — Manishearth ♦ Nov 15 '13 at 16:24
 
10:23 PM
Promoting the integral expression for the factorial of an integer
\begin{align}
n! = \int_0^{\infty} e^{-x} x^n dx
\end{align}
to the Gamma function (with $x$ complex)
\begin{align}
\Gamma(s) = \int_0^{\infty} e^{-x} x^{s-1} dx
\end{align}
we expect to be able to find an expression for $n^s$ by sending $x \to nx$ since
\begin{align}
\Gamma(s) = \int_0^{\infty} e^{-nx} n^{s-1} x^{s-1} dn x
\end{align}
gives the expression
\begin{align}
\dfrac{\Gamma(s)}{n^s} = \int_0^{\infty} e^{-nx} x^{s-1} d x
Obviously related to the Riemann Zeta paper claymath.org/sites/default/files/ezeta.pdf
 
10:44 PM
@DavidZ no, the policy decision previous to that one. If I understood the timeline correctly, they were initially on-topic, and then there was (when?) a decision to blanket-ban all list questions, which was subsequently turned over into some-restricted-questions allowed following this and this threads.
I'm looking for (a) confirmation that that timeline is accurate (or a correction if it isn't), and (b) a meta post (or equivalent) that spells out the transition from on-topic to the blanket ban, particularly as regards (b1) the timing of that transition and (b2) the arguments laid down for why that should be done.
@Blue No. That's from after this answer. There @dmckee states
> like I did during the period when books was on-topic.
which implies that book questions were considered off-topic on July 2013 but had previously been on-topic.
 
@EmilioPisanty ah, OK I see what you mean. Unfortunately I don't remember the timeline in any detail. But I don't remember much discussion about book/resource recommendation questions happening anywhere other than meta, so theoretically it should be there.
 
@DavidZ yeah, I figured as much, but my searches came up empty
they're certainly not tagged [books], there's very little with that tag pre-2013
it's just figuring out a way to fish them out
 
All I can think of to find them would be sorting by date and browsing, which is pretty inefficient.
 
with the added inconvenience that the SE "advanced" search doesn't seem to have selection by date ¬¬
 
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