23% Beyond the Riemann Hypothesis - Numberphile
Featuring Jared Duker Lichtman. More links & stuff in full description below ↓↓↓
Read more about this: www.maths.ox.ac.uk/node/65844
Jared Duker Lichtman: web.stanford.edu/~jdl18/
More videos:
Riemann Hypothesis - • Riemann Hypothesis - N...
The Key to the Riemann Hypothesis - • The Key to the Riemann...
Primes and Primitive Sets (an earlier video with Jared) - • Primes and Primitive S...
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This reminded me of the Anchorman quote "60% of the time it works every time"
So this is a sort of Parker Reimann Hypothesis?
Parker meme
Huh?
The Riemann-Parker Postulate?
Hilarious😂
In the same way that "Reimann" is a Parker spelling of "Riemann".
6:15 The reason π is used for pi, is because the word circumference in greek is περίμετρος which has the first letter π. The reason π(x) is used is because the prime numbers in greek are called πρώτοι αριθμοί which litterally translates to first numbers. Yet again π is the first letter. Coincidental that the letter is the same.
For those who don't read greek: the first one is "perimetros" and the second one "protoi aritmoi".
I thought it’s a capital pi that’s used for primes.
That’s for series multiplication like how sigma is to sum
@@theemperor-wh40k18 those who don't speak greek wouldn't know that "oi" is pronounced "i"
@@soupisfornoobs4081Not sure that’s really all that important, is it? would in any event have been pronounced /oi/ at some earlier stage of Ancient Greek, even though it has been pronounced /i/ for a long time now.
So many strange things tie back to the Riemann hypothesis. It's fascinating. I'm glad he took the time to explain it so clearly.
RIEMANN HYPOTHESIS
I think I gained a 23% increase in understanding of the Reimann Hypothesis. Thank You
More like 0.23%
Brady being a questioning viewer is such a good device for information
I can tell that this guy is IN THE ZONE at the moment. I love when I'm like that and my topic of focus is so clear. It's just the looming wipeout of depression that comes later that wrecks me.
Oof, I feel that to my bones. The grad school depression and burnout is REAL
Same
Need to socialise 😅
I get what you mean 😢
Love the enthusiasm of this guy
Every sentence is like "i can expand on this for an hour", and it makes me want any one of those hours
True, but he speaks so fast in short bursts, he lost me after 3 minutes…
@@NLGeebeeyeah… hard to understand for a non-native speaker like myself…
@@NLGeebee Actually I watched this at 0.75x speed
Asperger’s syndrome or adderall
It's inspiring to see, hear, and experience the growth of this channel. I wouldn't be surprised if the people who eventually do solve the Reimann Hypothesis are huge fans! It's not just that, it's the quality has been so consistent. Thank you for this.
i really enjoy the longer videos with jared, his explanations are great.
I loved the way he explained it and a different aspect of it. Lots of people will tell you about a graph and a critical line and a complex plain, but that visual representation is way to complex.
Complex Plane* But yes.
Joke's on you mate, this guy's explanation was too complex for me as well.
Complex, heehee
@@nazgullinux6601 You're all imagining stuff
@@nazgullinux6601 These spellings are related, anyway, just like "sheer" and "shire", which also mean "plain" and "plane".
This guy sounds so excited while explaining mathematics like imagine as a kid you discover something and are eager to show that to your parents and friends and siblings....... This excitement in his voice is kinda interesting at the same time being kinda contagious too😊😊
I spent years figuring out what this video explained in a short amount of time. Really great video.
it never stops amazing me how the riemann hypothesis links to so many different things and all things related to prime numbers, which means it's related to basically all of mathematics.
That last 0.000000001% is a lot more important in math than it is in household cleaners.
Not really. They use i without it having any definable value.
but 0.00000000001% of te remaining germs will reproduce exponentially and grow back to the original quantity in log time!
Super clear explanation. This guy is awesome
This guy dodged brady's question "How do you measure that digits even-out" like 3 times, which was the thing I was most curious about . I feel like his explanations were just too superficial.
I'm wondering if the "real" explanation was lost in editing because it was deemed to hard for the average viewer to understand, but I'd have preferred to see something I didn't understand that the completely unsatisfying result that we got instead.
No he didn't, he mentioned it can be done in terms of the variance
Very happy to see Jared back again
It is so nice to have the words hypothesis and theorem used correctly on youtube...
@jash21222Eh, probably misuse of "theory" is being lumped in with it.
It's unfortunate that the words "theorem" and "theory" are so similar; the difference in meaning is quite large.
Mathematics has a different use of it than the (other) sciences.
Mathematics has a different use of it than the (other) sciences.
But that's just a hypothesis, a game hypothesis
Always nice to see some progress!
Jared is so enthusiastic and excited about this subject it practically radiates through the screen. Find someone who talks about you the same way Jared Lichtman talks about the Bombieri-Vinogradov theorem!
More of this guy please, he's great
I feel like I talk just like him a lot of the time. The little pauses while still decoding the concept in mine mind.
I love watching these videos and just letting my ’tism run wild
1:27 My favourite way to do maths! Think of a phenomenon, write a program to seek examples and write them, run it, read 'em, look for patterns, ...
This hypothesis is so far over my head that it's not even funny. But I'm glad people like him are able to grasp it.
I forgot what tab I was on and I thought this was a video about a talking parrot. I got very confused by your comment.
It's not hard really. it just means that whenever ζ(z)=0, then either z is a negative integer or z is on the critical line (z = it+1/2)
@@youtubepooppismo5284i think they might be talking more about the deep connections it implies
@@youtubepooppismo5284 I'm guessing the OP doesn't understand what that function really is. It has a simple formula for Re(z) > 1, but to understand how it's defined elsewhere you need to understand analytic continuation. From my understanding there is only one function on the complex plane that is analytic everywhere except at 1 and matches the formula where Re(z) > 1. And if the OP is reading this, "analytic" on the complex numbers means it's differentiable everywhere it's defined.
@@anticorncob6 Analytic doesn't mean it's differentiable everywhere, that would be holomorphic. Analytic is when its Taylor series locally converges. Althought analytic and holomorphic are equivalent on the complex plane so you can use them interchangably, but they have different meanings. You are correct in saying that there is a unique analytical continuation of the zeta function for Re(z)
love this guys new take on an old topic
I love how this guy always brings colorful metaphors
As Brady says, that was a unique angle/approach to the RH, which I had not seen before - I had often wondered if working in another base would throw up something more interesting - but as usual, "I've done so. Arne Saknussemm"
I HAVEN’T SEEN NUMBERPHILE IN SO LONGGGGG THIS IS BRINGING BACK MEMORIES 😭😭😭
Maybe subscribe to the channel then? They are posting regularly, this is your fault
11:38 Brady touches upon a very important (but seemingly innocuous) question about pure science and applied science. Engineering is also applied science in action. Any engineer will tell you the amount of assumptions, scenarios etc. that we 'limit' a case to just so that we get closer to the answer for that specific instance. Just a short example, we know that the earth is spherical, but all spheres are locally plane (flat) and that is why we make do with flat rulers to measure the length of a table for instance. Because for most ordinary measurements it is enough to assume that we are dealing with flat surfaces. So even though 99.9% leaves out a lot of numbers, but for a specific case 99.9% is as close to a perfect result one might get.
ive rewatched this one a few times already, it's teasingly deep and addictive
when he says "a question comes up" and his face lights up i know im about to have my mind blown. i never got to see this side of math in chemistry
Finally, a video with Riemann Hypothesis in the title that I can understand.
I still don't entirely understand how "closeness" or "convergence" was measured in this sense, but the video was still fascinating to watch!
I felt like he kept dodging your Q about "what does it mean to even out" -- did you ever get an answer to it? or anyone here know? :) I guess if you relax "how even it needs to be", you could probably "get there faster" in a sort of trivial/definitional manner? All that to say, it was a great Q, but you should have kept pushing for an answer! :P
"Leveling out" is meant in the limit as x goes to infinity - do the number of primes with each allowed digit converge to the expected mean value? The point is that you allow b to grow as well, not just x. To illustrate an extreme example, suppose you consider all the primes up to b/2 in base b (eg all the primes up to 500 in base 1000). No matter what b you pick, the result is very far from being uniform across all the allowed digits: the larger half of your digits are never reached at all! Taking b to infinity doesn't help: no matter what b you pick, half of your digits will get nothing. A more interesting example is considering all the primes up to b in base b. There still just aren't enough primes to go around; even if you take b to infinity, you'll always find many "allowed" digits with zero primes having that digit (never getting anywhere close to the expected average) But if you consider way more primes, say all primes up to 10^b in base b, and you take b to infinity, then the proportion of primes with each last digit DOES converge to the expected mean. The digits themselves are changing because you're changing the base b; I'm saying that as you increase b, no matter what base-b digit you pick, the ratio between the expected average number of primes with that digit and the actual number gets closer to 1 the larger b gets. The Riemann hypothesis would say that you don't need nearly that many primes in order to see the convergence: just taking all the primes up to b^2 in base b would be enough. The 2023 result says the same thing happens if you only consider all primes up to b^1.63 - though this only holds for "most" b. That is, as you increase b, it's possible that for some very sparse sequence of bases, the distribution jumps away from being uniform; but as long as you take a sequence of bases b that avoids these rare troublemakers, then the proportion of primes up to b^1.63 with each allowed last digit converges to the expected value as b goes to infinity.
@@japanada11 I think you explained this part of it better than the video.
@@japanada11 that is definitely helpful, thanks for the details! Though I might still need to just read the papers to fully understand it. Based on their dialogue I was imagining the definition to be more focused on identifying some type of bound on "the variance across the digit-buckets" when you look at all (or "almost all") bases up to x^0.5 (or x^0.61, etc). Fun stuff!
I agree and I'd also like to know more about the local probabilities around each prime. it's nice to know when the probability evens out when considering the space of all primes. but are there some primes where certain last digits "disappear" for a while?
This is such an excellent video. Bravo
So thankful he took the time to show bases other than 10, much appreciated
Nice to watch and thanks for sharing
Jared is a good explainer. Obviously this is a subject thats far too complicated to explain comprehebsively, but he did a great job at giving an intuitive understanding of the core principle and offered glimpses of why this problem is so important to many different areas of maths.
amazing video, great explanation
One day this channel will reach ten million subscribers. I'm calling it.
This guy seems very nervous and extremely confident at the exact same time. Also very ackward and very elloquent at the same time. Very interesting.
Crazy how so much is connected to the Reimann hypothesis
Use the Riemann hypothesis to clean the house :D
J
When he said James Maynard was “up there “ I thought he meant he passed away😅
Thankfully not! I thought he meant James was watching over this presentation because of his recent Field's Medal award ... but I think James is humble and almost incapable of such hegemonistic thoughts.
As an economist, I was excited about Keynesian economics until I remembered that it is John, not James
13:58 AND THIS IS TO GO FURTHER BEYOND!
Jared's cadence reminds me of Christopher Walken. Lovely video :)
'Im not so fast at writing these numbers down... especially if they go on forever.' Yogi Berra-ism if Ive ever heard one.
Thank you for the video.
I love how Brady comes up with the example of the Fermat's Last Theorem - that a single counterexample would destroy it. Riemann Hypothesis is no different. But, from what I understand, Fermat's Last Theorem has now been proven. It's sort of a "race" between finding a single counterexample to the Riemann Hypothesis and finding a proof for it. I'm rooting for the proof, of course, but a counterexample would be fascinating.
I couldn’t help but notice, the number 0.617 is very close to the golden ratio that is approximately 0.618. I wonder if there is a connection 🤔
I actually doubt that. The golden ratio is the solution of the equation x²-x-1=0. You may find an infinite number of either functional results which may come close to that without having anything to do with it.
@@docwunderYet the golden ratio shows up in other places Say you wanted to solve the differential equation f'(x) =f^-1 (x), where f^-1 is the inverse function of x The solution involves x^phi
Funny how the length of the video 20:27 (2027) is also a Prime Number.
10:11 "Almost all" and "over 99.9%" are different; does anyone know which it is? In other words: as x → ∞, the percentage of bases b < √x for which (*) holds goes to some constant k; does the Bombieri-Vinogradov theorem state that k = 100% or that k > 99.9%? (The big-O notation on the wikipedia page for the B-V theorem makes me think it's the former, but I understood _almost none_ of that article :P)
Always good to see new mathematicians, I like this guy Base "e" - someone must have tried it, and I cannot believe the results were uninteresting.
If you follow the rule that all digits must be smaller than the base (so that the only digits allowed are 0, 1 and 2), integers above 2 don't even have a finite representation, so there's no last digit to look at.
I still don't understand the Riemann Hypothesis, but I'll try to understand it a bit better next time it's mentioned or explained in other videos.
I like this man!
The distribution of primes actually does have a connection to pi (or, if you prefer correctness over deeply-rooted tradition, to tau). 3blue1brown did an entire video on this connection.
The expressions 6n +/- 1 produce all prime numbers greater than three, and many more composite numbers. If we knew exactly where the composite numbers would appear in these sequences, we could infer the location of all of the prime numbers. Am I understanding this correctly? Of what use would this be to anyone?
Can you link me to the video you are referring to? I'm very curious about this video.
I like listening to Jared. I like his quantum speak.
He speaks in bursts.
😅😅😅
I love Mathematica for exactly this purpose
"I'm not buyin' it!". This is why we love Brady.
his enthusiasm is great
I loved the Seagull numbers!😂
I really like this dude. He could talk math to me all day
very cool result !! >:0
I still feel like I'm missing what "evenness" more precisely means here.
I believe it meant that the height of the bars becoming the same with small fluctuations in this case
@@tomalata5742 That's not very precise. How small are these fluctuations allowed to be? It's only 0 in the limit.
I still don't get what does 'starts to even out' mean... If you look at variance, what limit of variance do you set?
Yeah, he never explained what limit for "evening out" is being considered. I feel he should also have given some examples of an increasing b value as oppossed to just a constant base, since it seems hard to visualize.
I suppose infinite evening out. 0 variance as x tends to infinity.
This seems like a really neat result, but I'm afraid I'm not understanding what it's saying. Doesn't the "~" symbol in π(x; b, a) ~ π(x)/ϕ(b) imply that we're taking the limit as x -> infinity? (specifically, the limit of the ratio of both sides equals 1) So when we say b < x^(1/2), won't all x eventually satisfy that as we let x -> infinity?
He is saying that you can set b to sqrt(x) and it will still hold. You can let the base grow at the same time as the x
(from someone who doesn't understand much of what was said) it typically means 'proportional to', no?
The property you're looking for is "asymptotically equivalence", represented by "~". The definition is *"an ~ bn" "an / bn --> 1 for n --> oo"* Roughly speaking, if *"an ~ bn",* they have roughly the same behavior for large *n.*
From my understanding...Yes they will but that is the Riemann Hypothesis. The first is just the Prime Number Theorem, the RH simply asserts that we don't need to take limit to infinity but rather x only needs to be bigger than the square of b for the relation to hold
May your passion for mathmatics rub off in my life so I may find something im so passionat about, amazing lecture 🥳
Awesome video and this guy is explaining it very nicely. One small thing that I noticed is a short fraction of a second break he takes after every few words, anyone else noticed that? Again, great guy, nothing bad about it, but it just kept messing with my head for some reason lol
"Morally speaking..." I choke-coughed.
I had trouble following the connection with RH and the b < x^exp equations. For starters, if RH was proven, would that exponent become a new value? Like 1? Or are they measuring different things? Also, I don’t understand the concept of “beyond RH”.
The connection with the Riemann hypothesis is that it yields better approximations of the prime counting function. The generalized Riemann hypothesis then gives better approximations for Dirichlet L functiona which is what you want for this problem in particular. If you do some contour integration and use Perron's forumula, you can show that the Riemann hypothesis is equivalent to the bound psi(x) = x + O(sqrt(x)log(x)) which with some more integration can be shown is equivalent to the bound pi(x) = Li(x) + O(sqrt(x)log^2(x)). You get something similar for the Dirichlet L functions under the GRH which is what is really being used here.
The Riemann Hypotheses only gets you to 1/2 for this particular question. But if proven, it always works. There are other ways to get similar results, and actually stronger results (ie you get to evenness faster) but only work most of the time. These other ways have been proven.
@@billcook4768From my understanding the results are not necessarily stronger than the Riemann hypothesis but rather allow to you to do away with it and achieve the required result. Case in point here we see that RH asserts the statement is true for all b < sqrt(x) but then Bombieri-Vinogradov a slightly weaker result show that the statement holds for almost all b's which isn't equal to RH which says that it holds for all b, but this result allows us to do away with RH in some of the cases. Even the subsequent statements are just but the tightening of Bombieri-Vinogradov theorem
Very Interesting!
He never defined "evens out", he just said it's about the standard deviation. Be a bit more formal next time maybe? Also he's waving hands a lot and not explaining how the Riemann Hypothesis based result works, or any other results. Also not being able to give examples of bases that work for b < x^0.617 makes it even more vague. Maybe we proved that it works for all b < x^0.617, pushing the upper bound for b a bit beyond what we could prove using RH? I'm left with a bit of confusion on what exactly is the result that I should be excited about.
i watched the 9,1 sequence video and i can't unsee it (the 9,1)
This one I could follow, but it wavers back and forth about being specific. "Kind of nice even distributions" versus "we want a complete understanding". There's this handwaving feel about how meticulous you are in the distribution being equal, versus the tiny incremental testing of x.617. It sounds like digging to be exact while handwaving the parts that say you're exact.
Well that was very interesting and informative and very well presented.
a very beautiful handwriting he has
this dude could go on that for hours i can tell
No, this video is scripted. So no, you are wrong
ok@@ZelenoJabko
Great explanation!
Me at 2am not having taken a math class in a decade: hmm yes yes the Riemann Hypothesis of course
I'm not sure if this guy has ever been on the channel before but i like him!
What are the round things in the floor? Speakers? Ventilation? Access panels for power outlets and data ports? Surely not water drains.
This dude is great
My request for an insight into What is the link between Primes and zeroes of Riemann zeta function? I tried to learn on my own and I guess I am not smart enough.
finally Mathematica came into play!
It's not generally appreciated how many theorems in number theory (which deals with integers) are statistical (and thus involving real numbers). The Prime Number Theorem in its simplest form is an example.
My first question is what are the characteristics of that 0.1%?
The 0.1% is not proven. It MIGHT follow Riemann, or it might be different and break.
My understanding is that it’s not a 99.9-0.1 split or something like that, it’s an “almost all”-“almost none” split where the probability of it working is exactly 1, but there may still be failure cases. For example, it could fail for the primes, or for the powers of 2, or for the set {1, 50000000}, or never. The point is that we know it works “every” time.
The 0.1% was him trying to be a bit friendly (at least to pass his point) but when the technical word "almost" is used in Mathematics it simply means the probability of finding anything that violates what was being discussed is zero. It is just that when talking about probabilities (or any measure at that) of infinite sets a probability of zero does not mean impossible just improbable
@@tomalata5742 I'm not sure assigning a probability here would make sense (without choosing some arbitrary distribution for different bases). I think what is meant here is that when you look at bases up to a limit and that limit grows then the share of the bases where the hypothesis doesn't hold tends to zero.
@@seneca983 Hello I'm finding trouble following. Kindly elaborate. From my understanding sometimes we can make general statement about distributions without knowing the particulars of distribution for example, the chebyshev's inequality. we know it holds for a class of distributions that satisfy certain properties, the same could apply for this one, that is, the statement holds for the class of distributions that model the distribution of primes under different bases
Have you ever had someone talk about prime numbers in other bases? He touched on it here, but it got me wondering. I'd be looking for a pretty simple "Intro to other base primes".
A prime in any other base is still a prime.
@@davidgustavsson4000 Hmmm. Yeah I guess 5 is still 5, it just looks different. A bit tough to get my head around. I'm not a mathematician, can you tell? 😉
@@BLenz-114 for example, 111 base 2 (= 7 base X) is prime, because it doesn't matter which base you express it in.
Any prime isn't divisible by any number other than itself and one, so setting the base to any number won't change that fact.
Why does it seem like the limit is going towards x^0.618 (1/phi)? Edit: Timestamp 16:07 is where I'm talking about
I Have solved the Riemann hypothesis 8 years ago. I've reached out to many academics. NONE ANSWERED. Reached to this channel (multiple hosts) NONE ANSWERED. So I enjoyed the result and kept it to myself. ABSOLUTELY SERIOUS HERE. NO JOKES!!!
Proof?
@@rand0m_694 hint rather: the reason no one could solve it is because everyone is still using Euclid's 5000 year old primarility check formula. I found a better one and it opened a sea of new mathematics for me.
He has the proof, but it’s too large to fit in the comment box.
@@MrM1729 if you believe that the proof to one of Math's hardest problems should be posted in the comments, then I should believe your entire knowledge base should fit inside these comments.
I find that one of the fun things about primes is that there are all these hard and fast rules about them, except for the single digit numbers. It just seems cheeky that 2 and 5 sneak in there at the beginning.
That's because the thing that makes primes special is that every single one of them introduces a new rule that only it breaks. No prime is divisible by 2, except 2. No prime is divisible by 5, except 5. No prime is divisible by 23, except 23. And so on for every prime. It just happens that 2 and 5 are the ones that are visible in the last digit because they're the factors of 10 which we chose as our base to write numbers in.
Even 3 isn't THAT prime. If you look at a chart of primes in base 10, the ones ending in 3 go prime, prime, composite, prime, prime, composite, except the very first one "3" which breaks the pattern and is prime.
Haha what kind of rules are you using? 0,1,2,3,4,5,6,7,8,9 is manmade single digits. Let’s say we had one more digit for ten, T. In this system, T is ten, 10 is eleven, …, 19 is twenty, 1T is twenty-one and so on. Then these are the primes in Base-11. 2,3,5,7,10,12,16,18,21,27,29,34,38,3T,43,49,54,56,61,65,… For comparison, these are the primes in Base-10. {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,…} Notice 65 is a prime in base-11. That is 6*11+5 =71 in base 10. So, there’s nothing special about 2 or 5.
@@GynxShinx That pattern gets broken up by more and more composites as the numbers get bigger; you just might not notice it in reasonably-sized tables.
@@Jesin00 Yeah, still funky though. I feel like in another universe where 3 was larger, whatever that means, it may have been composite, where 7 on the other hand just seems very prime, even in other bases.
Jared is wonderful. More of him please😊
If the √x result links back to the zeros along the critical line of 1/2, what does this x^.617 result say about RH (if anything)?
So part of the reason this is tricky to answer is that the relationship is not actually to non-trivial zeros of the zeta function, but to non-trivial zeros of a more general class of functions known as L-functions (this is the "generalized" part of the "Generalized Riemann Hypothesis"). Now, the x^1/2 bound seen here is directly tied to the conjecture that non-trivial zeros of l-functions lie on the critical line Re(z) = 1/2, just as in the Riemann Hypothesis. Very loosely speaking (I don't understand the details myself), the Bombieri-Vinogradov theorem can be viewed as saying something about how the zeros of different L-functions interact, rather than saying something stronger about just one L-function. That's why it's a bit "orthogonal" to the GRH: it is in one way stronger by saying things about multiple functions at once, but weaker in terms of what it's claiming about those functions - namely, just some amount of cancellation between their zeros, rather than restraining them all to a line.
@@lppunto at this point I feel like mathematics approaches the realm of magic tbh
What are the practical applications that he is takling about? 17:21
Brady tried to ask at 7:28 and again at 14:28 and 16:08, but I never heard a clear answer about what "evening out SOON" means. I get that lim_(x→∞) π(x;a,b) / (π(x)/φ(b)) = 1, but it doesn't make any sense to me to say that a statement about limits "holds once x > b²". And the versions with "b < √x" or "b < x⁰ᐧ⁶¹⁷" make even less sense to me because if we stop at some finite x, then π(x;a,b) won't actually be π(x)/φ(b), regardless of b. I still have no idea what the actual claim about "closeness" is.
@9:41: OOOOOHHHH!! Dang! Wow... it's like finding Pi in continuing fractions a la Ramanujan, or in disparate, unexpected physical processes and... Then... wait... I still can't see Riemann clearly.
QUESTION: Is the 1985 finding supposed to actually be exactly x^(pi/6)?
I like how he is explanatig this!
I think that the Riemann hypothesis critical strip is related to the Central Limit Theorem or the "law of large numbers"? Why are the values of the z table or z scores of a all converging to 0.5 or 1/2... it's just something I've observed. I'm referring to the normal distribution or Gaussian distribution or the probability density function, by the way. :)
Never mind, somebody already noticed it -- "Large deviations of Selberg’s central limit theorem" by Louis-Pierre Arguin and Emma Bailey
Jared is a gem!!