@@AgentM124 the MathWorld article linked in the video description mentions that the bound's eventually broken in both directions, but not much else, so I'd guess it isn't known which direction is broken first.
@@5astelija75 It is. 10^40 = 10*...*10 40 times, so 0.5*10^40 = 0.5*10*10^39 = 5*10^39. Since 10^0.7 is roughly equal to 5 we can also write this as 10^0.7*10^39 or simply 10^39.7
@@5astelija75 10^39.7 = 5.01187...x 10^39, so you're right. In fact 10^(5.01 x 10^39) is 10^37 orders of magnitude larger than 10^(5 x 10^39). Big numbers are bizarre. What does the word "about" even mean any more
Just remember: the inverse of an implication is not equivalent to the original implication, i.e., the antecedent (Mertens conjecture) being false does not imply the consequent (Riemann hypothesis) being false. The Riemann hypothesis could still be true; we just don't get any help from here.
@@fllthdcrb why would you call it the inverse of an implication?..is that the right term?..wouldnt the negation or opposite be more correct..inverse is more like 3 vs 1 over 3 or reciprocal in math..think using that word is unclear..just saying..
@@leif1075 It's the _logical_ inverse. Totally different from a reciprocal (multiplicative inverse, which is part of algebra, not logic), and also totally different from a negation: Original implication: P → Q ⇔ ¬P ∨ Q Inverse: ¬P → ¬Q ⇔ P ∨ ¬Q ⇎ ¬P ∨ Q Negation: ¬(P → Q) ⇔ ¬(¬P ∨ Q) ⇔ P ∧ ¬Q If you're not familiar with the symbols, P and Q are statements, ∧ means "and", ∨ means "or", ¬ means "not", → is implication, and ⇔ is equivalence. The first equivalence on each line is by the definition of implication, and the last one on the third line is applying one of De Morgan's laws. Anyway, what I was getting at is, the inverse is not equivalent to the original implication, which you can see above. One implication using the same statements that is equivalent is the contrapositive: ¬Q → ¬P ⇔ Q ∨ ¬P ⇔ ¬P ∨ Q ⇔ P → Q.
I really want to know how it was proven that this number exists and breaks out of the parabola, since it is so big that we can never know what it actually is.
@@yeoman588 Not really an explanation but from a quick look at the paper, here's the idea: There's a thing in math called the limsup - if you know what a limit and a supremum are, then it's the limit as x goes to infinity of sup f(x). If you don't know what those are, it asks what's the highest value y of the function f so that no matter how big x is, there's a bigger x_0 with f(x_0) really close to that value y. In other words even when x is big, the function keeps wandering back to that maximum value. The paper looked at limsup M(x)/sqrt(x), the ratio between the sums of mu values in the video and the square root of x, and they found that limsup M(x)/sqrt(x)>1.06. In other words, M(x)>sqrt(x) infinitely often for large x. To show this limit, they needed to use a lot of computation with complex integrals which look yucky and I ain't gonna try to understand them :p . But that's math for ya!
Short version: find another function, which bounds the biggest values of M (involves Zeros of the Riemann ζ function) and then approximate this function. This is possible by knowing a lot (a few thousands) of zeros of ζ and it is a "nicer" function. Since this gets large enough sometimes, the conjecture is disproven.
The Pólya conjecture is similar, but you count DISTINCT prime factors (e.g. 10 and 20 both have two factors), and the conjecture is that the running total never goes above 0. It's true for a while, but it eventually fails, although at a more reasonable number: 906,150,257.
Because there's an infinite number of numbers from which to pick a new favorite number, so the probability of any particular number being Brady's favorite number is zero.
Well, that's what seem to happen in Mathematics, quite a lot. Think elliptical functions and modular forms. On the surface, no relation, until it was all tied up with Wiles proof (with a bit of help from others, of course) of FLT. Or go see 3blue1brown's video on colliding masses being a neat algorithm relating to Pi. Happens all over the place! That's all part of the fun!
Even weirder is how this conjecture if true would have proven the Riemann Hypothesis, which we really really think and hope is true. So the falsity of this is perhaps a little alarming and surprising.
@@homelessrobot "Imply" in the sense of logical implication, so A implies B means that if A is true, B is true. Not a colloquial English sense of "suggests" or "hints".
This is not the case here. The Mertens and Riemann conjectures are twins. The original question is whether there is an identifiable pattern in the distribution of primes among natural numbers. One research path led to creating the Zeta function and formulating the Riemann hypothesis about its zeroes. Another research path led to studying the prime decomposition of all numbers and formulating the Mertens conjecture. Both are just different attempts at answering the same initial question: "Is there a a way to instantly check if a number is prime or not".
Dr Krieger, also known as Mathematician Amy Adams. Also, perhaps I missed something, but why exactly does the function ignore (attribute zero) numbers with repeated prime factors?
By doing this you make sure that this is multiplicative, where if m has no common divisors to n and we let f(x) denote the function, then f(n*m)=f(n)*f(m). This might still sound a little arbitrary, but the function in this form pops up pretty naturally in a number of places like the mobius inversion formula.
I'm being serious, this is the most interesting thing I've ever learned. How have I not been aware that it breaks the sqrt barrier. I have a math degree and live in math, thank you Numberphile!!
@@samgraf7496 Both, according to the paper (first lines of page 3). M(n) goes above sqrt(n) and below -sqrt(n) for some values of n (infinitely many times). Don't know which one occurs first.
I love numbers. It's probably why in the middle of my college career I switched from engineering to math (and got a degree in it). I hated engineering but I loved all the math I was doing. I loved number theory but at that time it wasn't an area of math I could study, it was just a single class. Numberphile is so cool.
@Paul O'Reilly As a chemical engineer, I had to take calculus all the way through differential equations, then Engineering Math which is specific extensions of those areas, and also probability and statistics. Differential equations are pretty much the heart of engineering, because most things that matter in engineering problems revolve around change. Check out 3blue1brown's growing series on DEs to get a sense for this. All chemical engineering classes revolve around DEs. ChemEs, more that any other discipline are familiar with the Navier-Stokes equations (see the Numberphile playlist on these.) Now, as an engineer, I don't directly solve DEs...well, basically ever. But that's because most of that work has already been done or is underpinning the tools and equations and software we use on a daily basis. Engineering is about optimization, not exact answers. Not that there's anything wrong with that; nearly everything manmade you see right now in front of your eyes wherever you are is the result of engineers. We have to fold all considerations together to come up with an optimal design, because there's no perfect design. But math still underpins it all, because math is the language of physics, chemistry, biology, economics, and much more, and those are all the legs that the engineer's table is built on.
I am a Medical Student here but I love to Play with Numbers because it helps me to relax. I was trying to make Cross Product of Vectors easy as it was a nightmare in 12th Standard Maths and Physics I self discovered mu ijk in different form later I saw a video from Andrew Dawtson in RUclips to confirm if anything like that is there or Not. Yep it is there it was called something but it was epsilon ijk made me happy. Small ideas for fun
@Dr Deuteron Engineering is the science of understanding the high complexity of exact mathematical equations and approximating them with much more simple equations that are practical to compute in reasonable time and still very close to the hard real stuff.
@Dr Deuteron Electrical Engineering involves fourier series, differential equations, linear algebra, and more. Control engineering (a sub field) is almost entirely math that involves lots of calculus.
Actually that part was slightly misleading, as a weaker conjecture, not disproved by the counterexample to Mertens's conjecture, suffices for the Riemann hypothesis. (Basically the sum doesn't have to be smaller than the square root -- there's a bit of extra elbow room.) So the counterexample is very interesting, but not a tragedy for number theorists.
@@TimothyGowers0 If a weaker conjecture would have also proven Riemann's, then it's not misleading to say Merten's conjecture would have proven Riemann's. So, at the end of the day, Merten's Outlaw is still a disappointment (though neat to discover).
Yes, it is physically quite feasible. To the nearest order of magnitude, Avogadro's number is 10^29, so 1 gram of hydrogen has something approaching 10^29 atoms (and half that number of molecules). So 1 tonne of hydrogen has about 10^35 atoms, and 10,000 tonnes would be 10^40. If we freeze the hydrogen to keep the atoms in position, it occupies about 10 times the volume of the same mass of water, i.e. about 100,000 cubic metres, comparable to a large warehouse, hangar or conference centre. If each hydrogen atom can be in any of 10 different states, that's enough to encode our 10^10^40-digit number. Water ice might be more practical, which means about twice the volume and twenty times the mass. If we conservatively allow only two states per molecule (e.g. two isotopes of oxygen), we need 3 and a bit times more storage altogether. Adding some redundancy for error correction, we could still store the number - if we knew it - in a million-tonne block of ice 100 by 100 by 100 metres.
Great video! But 10^(10^40) has 10^40 decimal digits. So if we have 10^80 atoms in Universe, we have about 10^40 atoms for each digit in that number. So we CAN write it down if we really wanted
I feel like this is a glaringly unaddressed question, though it may take a few mathematical detours to answer (I have no idea): Why did Martens conjecture that the function was bounded by sqrt(n)? From the data we were shown, the function doesn't even seem to approach sqrt(n). Why not n^(1/3) or ... anything else? Very curious!
I'd love to see some kind of description of how the proof was done if that's at all possible. Also, do we know if it breaks the bound pisitively or negatively? Thanks for another great video!
Brilliant video. It has to be said that not all hope is lost proving the Riemann hypothesis through some statement on these numbers. Just tweak the conjecture a little.
@@yareyaredaze9450 I would have wished for a small remark in the video itself. I'm far from being able to understand the proof after investing a few minutes to skim through the paper, but a few seconds in the video saying how it was achieved would have been greatly appreciated.
Odlyzko and a co-worker managed to prove that the limit of the supremum of Mertens function as x goes to infinity is greater than 1.06 which disproves Mertens conjecture. Their proof used extensive computations on the roots of the zeta function from which that number emerged.
Imagine asking for a proof of this, when there's a proof in the description of the video. Unfortunately if you don't understand it, you simply don't understand it. It's not really possible to sum up a 32 page proof for a youtube comment section. It involves a lot of complex mathematics and the more someone explains to you, the more questions you would have.
Wait but if there are 10^80 atoms wouldn't it be possible to represent numbers up to 2^(10^80) using binary (eg. Divide universe into grid spaces, atom =1, no atom = 0)? And then 2^(10^80) = e^(ln(2)*10^80) is definitely > than 10^(10^40) ° e^(ln(10)*10^40)... So the number CAN be represented! It would take on the order of 10^40 or 10^41 atoms to do it...
7:25 "Does it just break away from that square root limit or does it blast past it?" Geometrically or arithmetically? Arithmetically, it blasts past it. Geometrically it goes at least 6% past it and probably more than that, all according to the paper you linked.
I wonder if the wiggles after that big number break away in both directions even further and further or if it calms down somewhere near TREE(3) or whatever.
@@danielroder830 From what was said in the video, it doesn't break away very far, though what "far" means when dealing with numbers of that size is debatable (and almost certainly not intuitive - work out what the square root of 10^10^40 is, and I think you may be surprised). FWIW, 10^10^40 is nowhere near TREE(3). It's not even anywhere near 3↑↑↑3.
@@danielroder830 What the paper says is that it continues to grow at a geometric rate that is about 6% bigger than the square root. It doesn't break away "suddenly" at 10^10^40; it just grows faster than the square root for large values of n.
The Mobius function was featured heavily in my number theory course in university. It's a rather interesting function because of the Mobius inversion formula.
I wish I had just one math teacher that could explain things half as well as Dr. Krieger does. Most math teachers I've ever had the displeasure of being taught by were nothing but a bunch of dimwits enamored with their own meaningless jabber. No wonder I thought I hated math.
If experiment was the only avenue available we would conclude with evidence far stronger than needed to convict in court or accept as theory in science that the conjecture is for all practical purposes true. And we'd be wrong. Is there a moral here?
So we know there exists a number n where |M(n)|>\sqrt{n}, but do we know whether M(n) is positive or negative there? I mean does the jagged graph break out of the bounding square root curves from the top or the bottom?
Thinking about Skewes' number, there really seems to be something about primes that makes them break our seemingly natural expectations if we dig really deep, i.e. look stupidly far.
What is the reason or logic behind the conjecture? Other than it would look neat? Lesson: because a theory is elegant or beautiful is no guide to it being true. Kepler found that hundreds of years ago.
@Gnomicality Maybe. It seems clear that the failure of the Mertens Conjecture does not DISPROVE the Riemann Hypothesis, or that would have been the big story.
@@christosvoskresye The Riemann hypothesis is "equivalent" to Mertens' function growing with exponent (1/2 + ε), rather than exponent 1/2 (which was Mertens' conjecture). So invalidating Mertens' conjecture is frustratingly close to disproving the RH, but only close... (as far as I know, it's not even known if there is a lower bound on the value of ε).
Maybe someone already recommended it, but I think we should call it Odlyzko-Riele Number. They are the ones that determined it's existence. Yes, I googled it.
This reminds me a lot of random walks and Brownian motion. Do we have any results for C*√n where C is any constant? What about C*n^(0.5 + Ɛ ) where Ɛ is positive and as small as we want?
I'm sure that's where the idea came from. On the other hand, look at that graph: it looks more structured than I would expect for flips of a fair coin. Of course, even a random walk has some chance of exceeding those bounds at some point.
@Mark W That got me thinking, what's the probability that a series of coin flips would exactly match that graph until it breaks out of bounds at 10^10^40? I wonder if that number is comparable to Graham's number
@@jacksonpercy8044 that probability is quite easy to work out. Let's say you want to copy this graph up to a point n (could be 10^10^40 or anything else) Then the probability that tossing a 3 sided coin/dice would produce the same graph up to that n is exactly (⅓)ⁿ. That's simply because for every 0≤k≤n you know exactly what the value should be so you have a chance of ⅓ that your coin matches that number
Couldn't we potentially write the number down though? Using the lower limit, in base 10 there would be 10^40 digits, which allows for 10^40 atoms per digit if we utilise the entire universe, which is actually quite generous.
Can we do anything with those numbers with repeated prime factors? Eg. could we mix the results of the function (-1,0,1) that 0 goes to even or odd number of factors and check how the function then behaves. Does itjust go to infinity or also jumps around 0?
Interesting. We could instead sum µ(rad(n)), the number of distinct prime factors of n. Or sum µ(sqf(n)), where sqf(n) is n's square-free part: the product of all n's prime factors p where p^2 does not divide n, so µ(sqf(n)) is the number of distinct non-repeating prime factors of n. Or sum µ(n/hsqf(n)) where sqf(n) is n's highest factor which is square -- I don't know if that is more pertinent.
@@rosiefay7283 I though of something similar. The issue is that all composite numbers are repeated prime factors so it won't hover around 0. If you do it with coprime numbers and primes it for sure won't hover around 0.
Dr Holly Krieger is my favorite presenter on numberphile! All the videos with her explaining the mandelbrot set are incredibly mindblowing and inspiring to me! This video was a wonderful treat to start my morning with, thanks!
The 10^(10^40) is only an upper bound on the first occurrence, there are no proven lower bound except the limit it has been tested to: 10^16. So it could be small enough to write down, though probably not very likely with such a large upper bound.
It's conceivable that the number is way smaller than the upper bound. Remember, it's still possible that the solution to the Graham's number problem is 13.
I am curious if all of this is related to the “law of large numbers” in some way. because there [if we consider the additive inverse (difference), and not the multiplicative inverse (quotient) of the numbers of positive and negativ outcomes] we have similar zigzag behaviour (naturally from the randomness), which zigzag is also mainly inside a parabola [+-sqrt(N), where N is the number of trials].
Dr Krieger is easily in my top three Numberphile experts. Simple explanations of complex problems that usually tend to be the type of math I’m interested in. Great video!
Excellent description of the Mertens Conjecture, and the counter-example found. Many thanks for the link to the mathematical paper disproving the Conjecture.
![Cody Johnson - I'm Gonna Love You (with Carrie Underwood) [Official Music Video]](http://i.ytimg.com/vi/yy9PuYMU29g/mqdefault.jpg)
More videos with Holly: bit.ly/HollyKrieger
Holly on the Numberphile Podcast: ruclips.net/video/QmfQQzjpdpM/видео.html
I didn't catch, but did they know anything about if it breaks the √n in the pos or the negative?
Fyi: I see "Mertern's Conjecture" instead of "Merten's Conjecture" in the RUclips description of this video.
@@AgentM124 the MathWorld article linked in the video description mentions that the bound's eventually broken in both directions, but not much else, so I'd guess it isn't known which direction is broken first.
Could you do a video about how this problem relates to the Riemann Hypothesis? I find the interconnections of mathematics to be really interesting
@@rob6129 Second that.
5:02 "Living around zero but in a really complicated way"... Wow, you just described my bank account
Lol
Stefan not so Reich
But you can be certain that at one point your money amount blows outside of the boundaries, it might be negative though.
HOLLLLAAAAA
@@Oblivion1407 What if my bank account is 5+3i? Would that be money that works not just across space, but also through time and/or dimensions?
The square root of 10^(10^40) is about 10^(10^39.7). Freaky how it's so much smaller but barely looks any different
Why isn't it 10^((10^40)*0.5) ? How does this math thing even work
@@5astelija75 It is. 10^40 = 10*...*10 40 times, so 0.5*10^40 = 0.5*10*10^39 = 5*10^39. Since 10^0.7 is roughly equal to 5 we can also write this as 10^0.7*10^39 or simply 10^39.7
To compare them properly - what power of 10 will give you 0.5? Take log (base 10) 0.5
@@5astelija75 10^39.7 = 5.01187...x 10^39, so you're right.
In fact 10^(5.01 x 10^39) is 10^37 orders of magnitude larger than 10^(5 x 10^39).
Big numbers are bizarre. What does the word "about" even mean any more
wow ok my mind is officially blown. restarting....
Dr. Holly one of the best.
Can Ali Tomruk
Or Isla Fisher
I know who Dr. Holly Krieger is, but I have no idea who Amy Adams or Isla Fisher are..
I'm perfectly happy with that.
@@mostlynothing8130 Pugsley Addams will play me in that movie.
pH7oslo “I’ve never heard of famous people. Aren’t I edgy and cool?”
dont you have world records on CTR?
"If this was true, it would imply the Riemann hypothesis!"
"It's false."
*uncomfortable digestive noises*
Just remember: the inverse of an implication is not equivalent to the original implication, i.e., the antecedent (Mertens conjecture) being false does not imply the consequent (Riemann hypothesis) being false. The Riemann hypothesis could still be true; we just don't get any help from here.
*jazz music stops*
Daniel Dawson I think the OP knows this. The discomfort comes from the fact that this doesn't help us with it
@@fllthdcrb why would you call it the inverse of an implication?..is that the right term?..wouldnt the negation or opposite be more correct..inverse is more like 3 vs 1 over 3 or reciprocal in math..think using that word is unclear..just saying..
@@leif1075 It's the _logical_ inverse. Totally different from a reciprocal (multiplicative inverse, which is part of algebra, not logic), and also totally different from a negation:
Original implication: P → Q ⇔ ¬P ∨ Q
Inverse: ¬P → ¬Q ⇔ P ∨ ¬Q ⇎ ¬P ∨ Q
Negation: ¬(P → Q) ⇔ ¬(¬P ∨ Q) ⇔ P ∧ ¬Q
If you're not familiar with the symbols, P and Q are statements, ∧ means "and", ∨ means "or", ¬ means "not", → is implication, and ⇔ is equivalence. The first equivalence on each line is by the definition of implication, and the last one on the third line is applying one of De Morgan's laws.
Anyway, what I was getting at is, the inverse is not equivalent to the original implication, which you can see above. One implication using the same statements that is equivalent is the contrapositive: ¬Q → ¬P ⇔ Q ∨ ¬P ⇔ ¬P ∨ Q ⇔ P → Q.
I really want to know how it was proven that this number exists and breaks out of the parabola, since it is so big that we can never know what it actually is.
Just read the paper, link is in the description. But first, in order to understand it, go get a PhD in mathematics.
@@w00tehpwn Ideally I'd like an explanation that _doesn't_ require a PhD to understand. 😅
@@yeoman588 Not really an explanation but from a quick look at the paper, here's the idea: There's a thing in math called the limsup - if you know what a limit and a supremum are, then it's the limit as x goes to infinity of sup f(x). If you don't know what those are, it asks what's the highest value y of the function f so that no matter how big x is, there's a bigger x_0 with f(x_0) really close to that value y. In other words even when x is big, the function keeps wandering back to that maximum value. The paper looked at limsup M(x)/sqrt(x), the ratio between the sums of mu values in the video and the square root of x, and they found that limsup M(x)/sqrt(x)>1.06. In other words, M(x)>sqrt(x) infinitely often for large x. To show this limit, they needed to use a lot of computation with complex integrals which look yucky and I ain't gonna try to understand them :p . But that's math for ya!
@@w00tehpwn But first, we need to talk about parallel universes
Short version: find another function, which bounds the biggest values of M (involves Zeros of the Riemann ζ function) and then approximate this function. This is possible by knowing a lot (a few thousands) of zeros of ζ and it is a "nicer" function. Since this gets large enough sometimes, the conjecture is disproven.
I love when Nicole Kidman explains maths to me.
Ha ha , I totally see that now
You mean Amy Adams?
ok mister killjoy
Holly is alot prettier than nicole kidman.
Robert Heikkilä exactly my thought
Holly's laugh makes my heart smile!
The Pólya conjecture is similar, but you count DISTINCT prime factors (e.g. 10 and 20 both have two factors), and the conjecture is that the running total never goes above 0. It's true for a while, but it eventually fails, although at a more reasonable number: 906,150,257.
2:08 "lets forget about zero and start with two. "
*sad one noises
I thought much the same, but in a less funny way, then I think I worked it out: 1 has exactly 0 prime factors, so it has an even number of them.
math people seem to forget 1 is a prime, its only factors are 1 and itself, 1.
@@pulsefel9210 One is not a prime. A prime number has exactly one factor, not including itself. 1 has zero factors, not including itself.
no primes can only be factored by multiplying itself by 1, so 1 fits since you cant multiply anything to get 1 except 1.
no primes are numbers with 2 factors
"You go to the gym to get in shape but what about your brain?"
Uh, I come here.
8:37 "That's my new favorite number." Why do I get the feeling Brady says this a lot?
only because he does
Because there's an infinite number of numbers from which to pick a new favorite number, so the probability of any particular number being Brady's favorite number is zero.
@@YtseFrobozz No, he doesn't pick them uniformly.
@@ShankarSivarajan I mean there's literally no way to pick natural numbers uniformly because of sigma-additivity of probability.
It's the name of the show.
I'm a simple person:
I see primes - I click like
I see Dr. Holly - I click like
.
.
.
I see Primes and Dr. Holly - I post a comment
Uladzislau Shulha We are on the same page
So, you UNclicked “like” second time you clicked it, then.
That really depends on defining "click like" as an event or command
@@alsorew No, the second like overflows into the comment section, since both likes and comments matter to the algorithm (or so I gather).
But you didn't post a comment to your own comment in which you saw both "prime" and "Dr.Holly"
One of the best channels on RUclips
thanks!!
@@numberphile Numberphile Deserve have 314,159,265,358,979,323 SUBSCRIBERS
@@EdbertWeisly or 2,718,281,828,459,045,235,360?
@@aradhya_purohit sure
Surely you've got to name it something awesome like Mertens' Nemesis if it's the first number to break the rule?
The Anti-Mertens Number.
A Mertensplex, perhaps?
brb writing this down on my math rock song name ideas list
Merten's Foil? How's that?
Mertens' Bane.
It's always crazy when one conjecture just happens to tie in another one when they seem to have nothing to do with one another.
Well, that's what seem to happen in Mathematics, quite a lot. Think elliptical functions and modular forms. On the surface, no relation, until it was all tied up with Wiles proof (with a bit of help from others, of course) of FLT. Or go see 3blue1brown's video on colliding masses being a neat algorithm relating to Pi. Happens all over the place!
That's all part of the fun!
but in this case zeta has a deep connection with primes and this Merten thing is built on the primes itself
Even weirder is how this conjecture if true would have proven the Riemann Hypothesis, which we really really think and hope is true. So the falsity of this is perhaps a little alarming and surprising.
@@homelessrobot "Imply" in the sense of logical implication, so A implies B means that if A is true, B is true. Not a colloquial English sense of "suggests" or "hints".
This is not the case here. The Mertens and Riemann conjectures are twins.
The original question is whether there is an identifiable pattern in the distribution of primes among natural numbers.
One research path led to creating the Zeta function and formulating the Riemann hypothesis about its zeroes.
Another research path led to studying the prime decomposition of all numbers and formulating the Mertens conjecture.
Both are just different attempts at answering the same initial question: "Is there a a way to instantly check if a number is prime or not".
Dr Krieger, also known as Mathematician Amy Adams.
Also, perhaps I missed something, but why exactly does the function ignore (attribute zero) numbers with repeated prime factors?
This is a matter of definition only.
The usual mathematical reason: because it's more interesting that way
What value would you assign to it then? 1, 0 and -1 are already taken. Plus or minus 1/2? And which one when?
By doing this you make sure that this is multiplicative, where if m has no common divisors to n and we let f(x) denote the function, then f(n*m)=f(n)*f(m). This might still sound a little arbitrary, but the function in this form pops up pretty naturally in a number of places like the mobius inversion formula.
@@sambachhuber9419 I see now. Thanks!
I'm being serious, this is the most interesting thing I've ever learned. How have I not been aware that it breaks the sqrt barrier. I have a math degree and live in math, thank you Numberphile!!
sarcasm, right?
@@thishandleistaken1011 Many of us have math degrees and PhDs and didn't know it, that's the point, it's a new discovery.
Gregory Fenn by new you mean 1985
It breaks the Squarrier
@@gregoryfenn1462 how did so many people manage to get PhDs without knowing they got them? /s
Proposal for the name of this number: Mertens downfall.
Edit: i really like all proposed alternatives in the replies!
Mertens' Bane ;-)
Odlyzko's Number named after the author who proved it
Mertens' Folly.
Merten's Conjecture First Counterexample (There could be more than one).
MCFC
@@samgraf7496 Both, according to the paper (first lines of page 3).
M(n) goes above sqrt(n) and below -sqrt(n) for some values of n (infinitely many times). Don't know which one occurs first.
Holly is my favorite Numberphile speaker. I am delighted to see her again, and it's primes too!
you need to get out more
yeah i clicked on this cause numbers or whatever
One question not addressed is, DOES IT BREAK AWAY UPWARDS OR DOWNWARDS??
From what I gather from other people who read the paper : both, infinitely many times.
As for the first break? No idea.
Picture showed upwards.
@@ubertoaster99 There was a large disclamer that the picture was an artistic rendition.
@@snbeast9545 The artist knew what they were doing. I'd bet on positive :)
If you’re referring to boundedness, then the authors of the paper say they think it’s not unlikely the limsup is infinite.
I love numbers. It's probably why in the middle of my college career I switched from engineering to math (and got a degree in it). I hated engineering but I loved all the math I was doing. I loved number theory but at that time it wasn't an area of math I could study, it was just a single class. Numberphile is so cool.
@Paul O'Reilly As a chemical engineer, I had to take calculus all the way through differential equations, then Engineering Math which is specific extensions of those areas, and also probability and statistics. Differential equations are pretty much the heart of engineering, because most things that matter in engineering problems revolve around change. Check out 3blue1brown's growing series on DEs to get a sense for this. All chemical engineering classes revolve around DEs. ChemEs, more that any other discipline are familiar with the Navier-Stokes equations (see the Numberphile playlist on these.)
Now, as an engineer, I don't directly solve DEs...well, basically ever. But that's because most of that work has already been done or is underpinning the tools and equations and software we use on a daily basis. Engineering is about optimization, not exact answers. Not that there's anything wrong with that; nearly everything manmade you see right now in front of your eyes wherever you are is the result of engineers. We have to fold all considerations together to come up with an optimal design, because there's no perfect design. But math still underpins it all, because math is the language of physics, chemistry, biology, economics, and much more, and those are all the legs that the engineer's table is built on.
I am a Medical Student here but I love to Play with Numbers because it helps me to relax.
I was trying to make Cross Product of Vectors easy as it was a nightmare in 12th Standard Maths and Physics I self discovered mu ijk in different form later I saw a video from Andrew Dawtson in RUclips to confirm if anything like that is there or Not. Yep it is there it was called something but it was epsilon ijk made me happy.
Small ideas for fun
@Dr Deuteron Engineering is the science of understanding the high complexity of exact mathematical equations and approximating them with much more simple equations that are practical to compute in reasonable time and still very close to the hard real stuff.
@Dr Deuteron Electrical Engineering involves fourier series, differential equations, linear algebra, and more. Control engineering (a sub field) is almost entirely math that involves lots of calculus.
That's all fine and dandy but what on earth is that design on that tele
we're gonna need a bigger universe!
Best comment
Is it possible that the first failure of Riemann's Hypothesis is a number as big as this one?
Yes, unfortunately.
It could be even bigger
it almost definitely is. no failures have ever been found
@@leofisher1280 We probably haven't looked quite that far, but yes, we've looked very far
You take that back
I'm thinking of a number between 1 and Tree(3).
With that link to the Riemann Hypothesis, I can almost hear the collective groan in the world of maths when this was disproven
Therefore the number should be called "Rie...maaaaan!"
No, it should be called the reeeeeeeee-mann
Actually that part was slightly misleading, as a weaker conjecture, not disproved by the counterexample to Mertens's conjecture, suffices for the Riemann hypothesis. (Basically the sum doesn't have to be smaller than the square root -- there's a bit of extra elbow room.) So the counterexample is very interesting, but not a tragedy for number theorists.
Still a great video though!
@@TimothyGowers0 If a weaker conjecture would have also proven Riemann's, then it's not misleading to say Merten's conjecture would have proven Riemann's. So, at the end of the day, Merten's Outlaw is still a disappointment (though neat to discover).
I was legit heartbroken when I found out this had wrecked a chance to finally prove the Riemann hypothesis.
I think that was the most interesting point made in the video. I didn't know that was under consideration.
10^10^40 has only 10^40 digits, so technically it's not impossible to write it down...
Yes, it is physically quite feasible.
To the nearest order of magnitude, Avogadro's number is 10^29, so 1 gram of hydrogen has something approaching 10^29 atoms (and half that number of molecules). So 1 tonne of hydrogen has about 10^35 atoms, and 10,000 tonnes would be 10^40. If we freeze the hydrogen to keep the atoms in position, it occupies about 10 times the volume of the same mass of water, i.e. about 100,000 cubic metres, comparable to a large warehouse, hangar or conference centre.
If each hydrogen atom can be in any of 10 different states, that's enough to encode our 10^10^40-digit number.
Water ice might be more practical, which means about twice the volume and twenty times the mass. If we conservatively allow only two states per molecule (e.g. two isotopes of oxygen), we need 3 and a bit times more storage altogether. Adding some redundancy for error correction, we could still store the number - if we knew it - in a million-tonne block of ice 100 by 100 by 100 metres.
Call it "Merten's Bane"
It's a big guy
@@faokie For you
Just imagine working with numbers so large. That you would need more mass of ink than there is mass in the universe to write them down
That problem then is unsolvable.
I think Ron Graham can relate to that
@@RB-jl8sm not really. Math is filled with numbers you can describe and not write down. E.g. all irrational numbers
@@BrosBrothersLP i see, so we dont need to imagine it because it exists anyway and hence it is not too interesting.
That was an unfunny joke.
It is intuitively clear (for a physicist) why the magnitude of M(n) should be about sqrt(n): it is similar to a random walk on the number line.
Taking it to be related to a random walk would imply the conjecture is false, by the law of the iterated logarithm (even after accounting for the 0s).
my typical numberphile viewing experience:
start of the video: **yawn**
end of the video: **screams geometrically**
then don't watch - dork
Great video! But 10^(10^40) has 10^40 decimal digits. So if we have 10^80 atoms in Universe, we have about 10^40 atoms for each digit in that number. So we CAN write it down if we really wanted
In fact seems like there are more than 10^40 atoms in a star. So we just need to use one.
When i see notification my heart goes +1-1+1-1+1-1
_It goes bom-bodi-bom-bodi bom-bodi-bom-bodi bom-bodi-bom-bodi bom_
_Goodness gracious me_
So your heart shrinks to half its size
@@andrewtan881 Nah its 0
@@andrewtan881 his sum evaluates to zero as he has considered only finitely many (6 to be precise) terms
Fair enough
More videos with Holly! Please more Holly!
I feel like this is a glaringly unaddressed question, though it may take a few mathematical detours to answer (I have no idea): Why did Martens conjecture that the function was bounded by sqrt(n)? From the data we were shown, the function doesn't even seem to approach sqrt(n). Why not n^(1/3) or ... anything else? Very curious!
It’s not quite sqrt (n) it’s most likely it’s n^1/2+epsilon
Iirc it's because sqrt(n) is how max(random walk) grows.
I'd love to see some kind of description of how the proof was done if that's at all possible. Also, do we know if it breaks the bound pisitively or negatively? Thanks for another great video!
They should name it Holly's Number.
Or Merten's outlaw
You win :)
Merten's outlaw is the best one I've seen
Upvote Merten's Outlaw
Or they could name it TWO (all caps).
@@christosvoskresye ...? why would they do that?
If Dr. Holly Krieger were my maths professor I would be in college right now
Why does everyone always blame their previous teacher on their own failure?
@@irwNd2 No one does, the only person to say so is you.
Smart and beautiful, wow.
0:00 If your date says this, congratulations.
Damn it, Brady! Now I'm forced to take a study break.
Incredible, love this video - the graph looks very similar to a Brownian motion with no drift.
Makes me think of a random walk.
Brilliant video. It has to be said that not all hope is lost proving the Riemann hypothesis through some statement on these numbers. Just tweak the conjecture a little.
How do we know that the Mertens Conjecture is not true? What gives us the 10^(10^40)?
@@yareyaredaze9450 I would have wished for a small remark in the video itself. I'm far from being able to understand the proof after investing a few minutes to skim through the paper, but a few seconds in the video saying how it was achieved would have been greatly appreciated.
Odlyzko and a co-worker managed to prove that the limit of the supremum of Mertens function as x goes to infinity is greater than 1.06 which disproves Mertens conjecture. Their proof used extensive computations on the roots of the zeta function from which that number emerged.
Keep in mind that 10^(10^40) is only a bound, not the exact number.
Imagine asking for a proof of this, when there's a proof in the description of the video.
Unfortunately if you don't understand it, you simply don't understand it. It's not really possible to sum up a 32 page proof for a youtube comment section. It involves a lot of complex mathematics and the more someone explains to you, the more questions you would have.
I'm a simple man.
I am a depressingly simple man
aren't we all ?
Such huge primes....
5:11
Wait but if there are 10^80 atoms wouldn't it be possible to represent numbers up to 2^(10^80) using binary (eg. Divide universe into grid spaces, atom =1, no atom = 0)?
And then 2^(10^80) = e^(ln(2)*10^80) is definitely > than 10^(10^40) ° e^(ln(10)*10^40)...
So the number CAN be represented! It would take on the order of 10^40 or 10^41 atoms to do it...
it sounds kinda lame that numbers with repeating factors get assigned a 0...
Can we please hold on for a minute an apprechiate that the video wasn‘t stretched to 10 minutes 🙏🏽❤️
7:25 "Does it just break away from that square root limit or does it blast past it?" Geometrically or arithmetically? Arithmetically, it blasts past it. Geometrically it goes at least 6% past it and probably more than that, all according to the paper you linked.
I wonder if the wiggles after that big number break away in both directions even further and further or if it calms down somewhere near TREE(3) or whatever.
@@danielroder830 From what was said in the video, it doesn't break away very far, though what "far" means when dealing with numbers of that size is debatable (and almost certainly not intuitive - work out what the square root of 10^10^40 is, and I think you may be surprised).
FWIW, 10^10^40 is nowhere near TREE(3). It's not even anywhere near 3↑↑↑3.
@@dlevi67 AFTER that number, it breaks away at 10^10^40 and after that, i wonder what happens after that.
@@danielroder830 What the paper says is that it continues to grow at a geometric rate that is about 6% bigger than the square root. It doesn't break away "suddenly" at 10^10^40; it just grows faster than the square root for large values of n.
+
6:50 STONKS
The Mobius function was featured heavily in my number theory course in university. It's a rather interesting function because of the Mobius inversion formula.
If Holly had been my professor in collete I would have actually attended Maths classes.
lol
~just replied to get notified for future comments~
@@andermium ikr?
I would have gone to office hours
I think I attended most math classes. Because you miss one math class, and you are seriously behind in many cases. It's a pyramid scam! :)
Thumbnail looks like they are talking about Reddit voting system
I wish I had just one math teacher that could explain things half as well as Dr. Krieger does. Most math teachers I've ever had the displeasure of being taught by were nothing but a bunch of dimwits enamored with their own meaningless jabber. No wonder I thought I hated math.
Brilliant quote there
I love math but I wish I had Cliff Stoll as a math teacher, the enthusiasm he has for math is contagious.
@@Kraenesk I remember Cliff from his book the Cuckoo's Egg I think.
If experiment was the only avenue available we would conclude with evidence far stronger than needed to convict in court or accept as theory in science that the conjecture is for all practical purposes true.
And we'd be wrong.
Is there a moral here?
PifflePrattle I think it’s fair to question the utility of a conjecture whose truth depends on a quantity we could never compute.
„maybe you go to the gym [...]“ I‘m watching math videos for fun who you think I am
Hey Prime Numbers aren't THEE Only thing thst Soo Sizzling RIGHT NOW LASS!
So we know there exists a number n where |M(n)|>\sqrt{n}, but do we know whether M(n) is positive or negative there? I mean does the jagged graph break out of the bounding square root curves from the top or the bottom?
both things happen infinitely often
actually, the state of the art result is that it happens infinitely often that M(n)>1.82*\sqrt{n} and M(n)< -1.83*\sqrt{n}
Awesome! Thanks!
I bet Ramanujan would have answered this question
Love Dr Holly !!!
Merton’s Conjecture first failure is my favorite number and Dr. Krieger is my favorite mathematician
Thinking about Skewes' number, there really seems to be something about primes that makes them break our seemingly natural expectations if we dig really deep, i.e. look stupidly far.
0:59 of course they won't yell at you, because the function of mu is to mute them.
More Dr Krieger please!
I'm all for calling it Brady's number. Puts Google to shame.
wait waitwait, there is letter "meow"? ;)
"mu", the Greek m
What is the reason or logic behind the conjecture? Other than it would look neat? Lesson: because a theory is elegant or beautiful is no guide to it being true. Kepler found that hundreds of years ago.
I love how excited Dr. Krieger is about this one! excellent stuff thank you!!!
RUclips Algorithm... How did I end up here?
I wish he had asked if the sum exceeds the bounds infinitely many times.
@Gnomicality Maybe. It seems clear that the failure of the Mertens Conjecture does not DISPROVE the Riemann Hypothesis, or that would have been the big story.
@@christosvoskresye The Riemann hypothesis is "equivalent" to Mertens' function growing with exponent (1/2 + ε), rather than exponent 1/2 (which was Mertens' conjecture). So invalidating Mertens' conjecture is frustratingly close to disproving the RH, but only close... (as far as I know, it's not even known if there is a lower bound on the value of ε).
It does.
Maybe someone already recommended it, but I think we should call it Odlyzko-Riele Number. They are the ones that determined it's existence. Yes, I googled it.
This reminds me a lot of random walks and Brownian motion. Do we have any results for C*√n where C is any constant? What about C*n^(0.5 + Ɛ ) where Ɛ is positive and as small as we want?
I'm sure that's where the idea came from. On the other hand, look at that graph: it looks more structured than I would expect for flips of a fair coin.
Of course, even a random walk has some chance of exceeding those bounds at some point.
@@christosvoskresye Given enough time, random flips of a three sided coin will produce this graph infinitely many times.
@@MagruderSpoots That would produce any graph though.
@Mark W
That got me thinking, what's the probability that a series of coin flips would exactly match that graph until it breaks out of bounds at 10^10^40? I wonder if that number is comparable to Graham's number
@@jacksonpercy8044 that probability is quite easy to work out.
Let's say you want to copy this graph up to a point n (could be 10^10^40 or anything else)
Then the probability that tossing a 3 sided coin/dice would produce the same graph up to that n is exactly (⅓)ⁿ.
That's simply because for every 0≤k≤n you know exactly what the value should be so you have a chance of ⅓ that your coin matches that number
Hmm, I wonder if the mu function is breaks the square root bound on the positive or the negative side at the first counter-example.
When the content is soo good, I put the effort to watch every second of the ad that you so considerately put at the end of the video.... Cheers!
Couldn't we potentially write the number down though? Using the lower limit, in base 10 there would be 10^40 digits, which allows for 10^40 atoms per digit if we utilise the entire universe, which is actually quite generous.
Can we do anything with those numbers with repeated prime factors? Eg. could we mix the results of the function (-1,0,1) that 0 goes to even or odd number of factors and check how the function then behaves. Does itjust go to infinity or also jumps around 0?
Interesting. We could instead sum µ(rad(n)), the number of distinct prime factors of n. Or sum µ(sqf(n)), where sqf(n) is n's square-free part: the product of all n's prime factors p where p^2 does not divide n, so µ(sqf(n)) is the number of distinct non-repeating prime factors of n. Or sum µ(n/hsqf(n)) where sqf(n) is n's highest factor which is square -- I don't know if that is more pertinent.
@@rosiefay7283 I though of something similar. The issue is that all composite numbers are repeated prime factors so it won't hover around 0. If you do it with coprime numbers and primes it for sure won't hover around 0.
We should definitely name that number...
I vote for Mertens Bane
Dr Holly Krieger is my favorite presenter on numberphile! All the videos with her explaining the mandelbrot set are incredibly mindblowing and inspiring to me! This video was a wonderful treat to start my morning with, thanks!
5:55 very interesting entity
Holly
Also Holly *This implies the Riemann hypothesis*
Why does 10 have 2 primes (2, 5) excluding 10 itself, but 5 has 1 prime (5)?
Because 10 isn't a prime
Got it. "Prime" factors ;)
Dr. Holly Krieger the beauty of mathematics! :)
7:15 ❤️
The 10^(10^40) is only an upper bound on the first occurrence, there are no proven lower bound except the limit it has been tested to: 10^16.
So it could be small enough to write down, though probably not very likely with such a large upper bound.
It's conceivable that the number is way smaller than the upper bound. Remember, it's still possible that the solution to the Graham's number problem is 13.
I love your performance in Arrival!
I came to the comment section just to see someone write this comment!
Oh, not 10^400... 10^10000000000000000000000000000000000000000. Gotcha. *sulks away in existential fear*
0:20 ｂａｎａｎａ
I am curious if all of this is related to the “law of large numbers” in some way. because there [if we consider the additive inverse (difference), and not the multiplicative inverse (quotient) of the numbers of positive and negativ outcomes] we have similar zigzag behaviour (naturally from the randomness), which zigzag is also mainly inside a parabola [+-sqrt(N), where N is the number of trials].
Dr Krieger is easily in my top three Numberphile experts. Simple explanations of complex problems that usually tend to be the type of math I’m interested in. Great video!
This is probably the smartest comment section on RUclips haha
Love Holly’s guest spots
this video relates to the 3blue 1Brown recent video on this channel
When you want to comment first but you have to watch the whole video.
I will edit this later anyways
why are you making even your atatements sound like a question. effing annoying
The connection with the RH is very interesting and I worked on that in an expository paper for my number theory course.
the Riemann Hypothesis is really everywhere.
So beautiful
Excellent description of the Mertens Conjecture, and the counter-example found. Many thanks for the link to the mathematical paper disproving the Conjecture.
Doctor Krieger ❤️❤️ mmmm
I find it difficult to concentrate on the subject sometimes...
@@Kris.G yep...two very big distractions
5:07 describes dating really well: complicated, a lot of cancellation and I never was too big xD