The hidden link between Prime Numbers and Euler's Number
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- Опубліковано 11 гру 2020
- We will discuss how miraculously Euler's Number appears when asking how many factors a number has on average, which is closely related to the distribution of prime numbers. I still remember how amazed I was, when I first learned about this fact, so I had to share it with the world.
- Наука та технологія
Another way to arrive at the same answer is to think that on average, n/1=n numbers are divisible by 1, n/2 are divisible by 2, n/3 by 3 etc. So the average number of divisors is (n+n/2+n/3+ ... + n/(n-1) + n/n)/n = 1+1/2+1/3+...+1/(n-1)+1/n which is the sum of the harmonic series up to n. With the same trick of the area under a hyperbole, it turns out this sum approaches ln(n) for large n.
thats exactly what we did in the video
@@debblez Maybe you watched another video? Read the comment again and compare with the video: not the same!
Great video, the sound should be a little louder as the volume of this video is low compared to other videos :)
Agree wholeheartedly
Indeed
It is a bit muddled. This does not help to hear through the accent.
112ndtlkr
Excellent content, but the background music makes it hard to follow, which is annoying. Not sure why so many UA-camrs feel the need for background music when doing voiceovers. You should edit it out and repost without losing views. There is of course a YT video for that.
Awesome video. A better average is log(x)+2c-1, where c is the Euler-Masceroni constant. You get this if you only integrate your curve up to sqrt(x), account for the symmetry of the curve, and use a better estimate for the harmonic sum. It gives you a much smaller error.
Thanks. When I saw the graph, my thought was immediately that it must be a contant to improve that estimation but I didn't know its value.
When you write log with no indication as to the base then the base is conventionally assumed to be 10. If the base is e then it is conventional to write ln. That has been conventional since before I was born, (I am a grandfather). Example I had an instructor who had a PhD in physics who followed that convention. It is also more efficient to follow that convention when writing.
@@dannygjk It depends on the field. For example, in computer science with big O notation, it’s convention to leave off the base with an understanding that it can be taken to be base 2. I imagine in most science classes or research they might make the distinction more clear, but they tend to use log to refer to base 10. However, in number theory (as this approximation is widely used in number theory) it’s common for log to mean ln, since that’s the most common logarithm we talk about.
When there are multiple conflicting bases, then we write the base or use ln. It’s all about clarity 😄
@@JM-us3fr I studied comp sci it could be any base if you only write log. Only in specific circumstances can you safely assume it's base 2. Comp sci is universal as far as bases is concerned just like math. In comp sci I typically used base 2, 10, and 16 ocassionally I used base e or 8. Even base 256 can be useful depending on what you are doing.
@@dannygjk Oh I'm sure you're right, I'm just giving my experience with notation when I learned computer science. It was an algorithms class, so because of the big O, the base of the logarithm often didn't matter. Either way, if you ever see Terence Tao right his natural logs, it's always log, and this is common in number theory.
Even though the error reduces gradually, it always looks like the averages are a constant distance from the logarithm curve, no matter how big the number. I noticed a comment below added, "A better average is log(x)+2c-1, where c is the Euler-Masceroni constant"
I’ve been hunting for an intuitive explanation for why e shows up in the distribution of primes. Your video has at long last given me what I’ve been searching for. Thank you!
Discarding one part of area and taking the other felt rather hand-wavy. Together with slowly converging numbers at the end it leaves to think there might be more accurate approximation.
I agree. That was bad maths. At least, they should have indicated that a rigorous proof does exist, but that it is outside the scope of this video to discuss it (though, it logically makes no sense for it to be outside the scope of the video, since it is literally the crux of the entire video).
Wow. I've never thought about the exp function like this before. They should teach this explanation in schools so people can actually understand what the exp and ln functions are.
You were never taught that e^x is the solution to the differential equation y = y' ? Even though that's the whole point of the function?
@@12-343 maybe it was introduced in such a way that you had no idea why that would be important or interesting, at the time it was introduced, lol.
once you had enough knowledge it know it was important and interesting, the nature of e has been forgotten.
@@12-343in middle and high school the goal is merely spreading the awareness that e and pi exist. Most people dont respect math and respect less numbers that arent intuitive or can be obviously and directly used on the farm, or in daily conversation. In this climate, spreading awareness of e is the main goal, and showing some of its magic. The most in-depth the typical high school experience gets is in banking and continuous interest. Seriously. Which makes sense, given most people wont respect math, science, or anything outside of local cultural norms unless they themselves are gonna use it. the question of maximizing compound investment to its limit spawns a natural placement for e. Basically, theyre trying to introduce it to where the characters in the . movie of mathematics arent speaking in the language of exposition. That instead its closer to how the original thinkers came up with this stuff, step by step, and often thinking about real world applications. (Yes not always, but it was rather necessary for comfortable income.) Anyway. Literally nothing about calculus, limits, derivatives were taught in the high school setting until at least 2015 or so, started with AP and (sometimes) gifted programs. Then trickled into the typical classroom. Remember teaching is a grand strategy, like a large-scale war. The thinkers at the state level have to reveal the strategy over time like playing chess. Except, its much more hostile. No matter how well you play the game, parents hate tf out of you always and the masses blame you for everything, and in yhe classroom and school level, theres so much breakdown of the vision that its all rent like a bunker after being hit by a bunker buster FAB. I mean, in 2009-2012, we literally had EVERY SINGLE Georgia performance standard in mathematics itemized down to every objective of every lesson and every angle of each objective on posters on the walls. For every year at the same time. They were THE opening of each lesson and chapter. The literal clear and simple performance standard and clarification of the objective in every possible corner more than ads in a news outlet webpage today, and there was still total chaos in implementation. In my school, the dude who wrote the state curriculum visited my gifted class and after 90 minutes said he himself had no idea what the teacher was teaching. And hes a master mathematician.
So to answer your question. No, and really, why would you expect anyone to know that unless they are a math enthusiast? They may know continuous interest, and thats it.
You can get a better bound on the error than assymptotic correctness by using the Euler-Mascheroni constant; the limiting difference between the harmonic sum and the natural logarithm (and it's not too hard to show that this limit exists).
This is an excellent video. Please, make many more of these!
Thanks a lot!
7:30 and 7:40 I know it's beyond the scope, but would be cool to see a proof of how this error goes to 0.
i second this
From the graph on the screen it certainly didn't appear to go anywhere near zero. Seemed to be off by a constant. Of course if you look at the relative error, a constant divided by a larger and larger number goes to zero, while you still have a constant absolute error.
The error doesn't approach zero, only the percent error
Error goes to γ (Gamma). It's the percent error that goes to zero. Percent error is:
100*(Real value-Approximation)/Approximation
Our approximation in ln(x), so we can rewrite this as:
100*(Real value-ln(x))/ln(x)
Since ln(x) approaches infinity as x goes to infinity, 100*(Real value-ln(x))/ln(x) goes to 0.
@@azfarahsan"i second this" what? Say it correctly. Not surprised to see only 2 likes.
Appreciation to you. This should be one of the most suggested videos
How does this only have 3000 views? This is extremely well done and underrated
Is 100k views enough? Looks like the algorithm picked it up after your comment!
Ayo that’s awesome congrats lol, glad to see this got more attention
There is another graph with the property of all derivatives and integrals being the same, it’s Y=Sin(x)^2 + Cos(x)^2 - 1
Great video. I have never quite grasped intuition for why the ln function and primes are linked. The lattice points and the n/x function made it simple to understand! Thank you.
I LOVE YOUR LOGO, I LITERALLY DREW THIS WHILE I was working at Mathnasium! 1/6 + 1/3 + 1/2 = 1!!!
amazing. you choose the best topics, and explain them beautifully.
Nice intro video that uses only basic highschool calc to derive the main term in the asymptotic expansion in an accessible and visual way. The content was engaging and got me into looking for more details about the finer points on the next order terms. Keep up the great work :)
Just ran into this video. Amazed by the thought! Thanks!
Gorgeous video. Bravo!
This is so beautiful, thank you so much for this.
Said thing this video was recommended to me only now
Great video, very informative
Hope to see some more from you in future
This reminds me of what Prof. Dunham wrote about in "Euler the Master of Us All", the relationship between ln and harmonic series, he worked on sum of 1/k, Mascheroni did introduce the symbol gamma, though he allegedly miscalculated it, then came the famous sum of 1/k^2, where the Bernoulli were stumped. Love the beautiful graphics, very educational.
Great video! Thank you
The video was uploaded a year ago, I hope that you’ll eventually upload more of them! I’ll be definitely waiting
Incredible! Bravo!
What?! I thought you must have like 100k subscribers before I saw you only had 2 videos. Please post videos more regularly, they are really good!
Wow, connecting the sum of divisors to the integral of the reciprocal is very intuitive but I never thought about it that way.
Great Video and pleasant voice and background music!
What a nice video, I hope you can make more in the future, it's a shame it didnt take off when you published it.
Superb explaination!
Excellently done, subscribing for sure!
Amazing content. Thank you.
So this mentions primes at the beginning, but goes on to only talk about counting divisors. What did I miss?
Beautiful!
Great video, keep it up !!
the constant difference between ln(x) and the graph appears to approach -0.1544313298...
or 1+2𝛾 where 𝛾=-0.5772156649... is the Euler-Mascheroni Constant
This was awesome 👌
My first idea after seeing the curve was that it looked like the natural logarithm. Funny how intuition can guide us to the solution
Beautiful.
6:36 so you could define primes as integers "a" such that the function a/x only intersects with the integer lattice at a,1 and 1,a?
The picture also shows that when you want to check if n is a prime you just have to check divisors up to n^(1/2)
Cool :)
Really impressive visualizations! And clearly explained as well, love it!
I had to turn on the captions to understand what he was saying, because of his strange accent. I think he might be a foreigner or something, unfortunately, but the video was pretty decent, though, other than that. I just hope he’s legal, at least, since I supported his content, by watching the video
Terrific video
More videos! Please! ❤️
Great vid!
i finaly learned wft the slope number means, thankyou
Music too distracting.
Good video ! But don’t we have some multiple of the Euler mascenori constant as the limit of the difference ? 7:50
We do, that’s why there’s ~ sign. He used percentage error - as ln(n) grows to infinity, the percentage error indeed tends to 0
I think the percentage goes to 0 but the average tends exactly to H(n) (nth harmonic number)
Really great video! I like the background music, though the overall volume of the video is a bit low
this was lovely. is it related to hardy-littlewood?
Post more! Great one though ❤
Congratulations, well done. The explanation is amazingly simple. I'll critic one thing (not very important;): the sound volume is low.
Superb.
I knew it. It's, like an onion, the deeper you peel it, the more it stinks.
Why does the percent error go to zero? How do you know that?
Please either change the title for this video or explain much more clearly the connection between Euler's number and prime numbers. I watched this twice, and enjoyed it, but I don't see what your title promised.
I keep forgetting it's "Oiler", not "Youler".
8:00 why exactly does the error not matter in this case? I feel that this is not immediately obvious and needs to be proven
Amazing video! Why is the average number of divisors equal to the number of primes?
7:39 I don't really understand this step. How do you know the first column ends up filling in the cracks of the area under the curve?
The integral would diverge without removing that column. That hyperbola goes up forever when approaching zero, as you take reciprocals of tiny numbers. It had to be done to avoid that inconvenience. And with filling up the spaces - the integral itself includes those, that’s how it works, but I think the area of those extra bits become insignificant compared to the squares as n increases.
It doesn’t fill in the cracks, or if it does that’s irrelevant. Both the first column and the cracks have an area which as a of the % of the total area tends to 0.
@@zildijannorbs5889 That's not how that works. First of all, there is nothing that even justifies taking the integral here.
@@angelmendez-rivera351 but there's clearly an integral in the video, right? I thought what I said makes sense.
Function for prime number check if natural number N is prime number is:
Π [j=2 to j=(N-1)] sin(π*N/j)=a
a=0 for not prime
a≠0 for a prime
if you know isin(x)+cos(x)=e^(ix) so that means that sin(x)=-i*(e^(ix)-cos(x))
so here you have link between euler number and prime numbers.
Great and interesting video. But why the area of left side equals the upper side area? Didn’t get explained l.😂
Can someone explain (or suggest a reference to read) regarding the relationship between average number of factors and the primes?
one question to ask: what is the difference between the number of factors and the approximation? i'm thinking this difference itself doesn't tend to 0, but tends to some other function
See several other comments - you can describe this difference by using the Euler Mascheroni constant.
The audio is so low that I had to put the headphones on, then the music didn't help because it would cover up your words. I want to watch, but it is hard to understand the audio.
i also like the clash of clans music in the background
If a(x) was the sum of averages of divisors. Then a(x) / x where x is total numbers is equal to ln(x) does it mean that this function a(x) equals to ln(x)*x
This felt like a light theme 3blue1brown video.
9:33 how cool. i never thought of inverse functions as swapping the axes.
(9:33 lol perfect-square timestamp)
Slope of ln (n) as n tends to infinity is zero. Doesn't this imply ln (n) is bounded above?
Ah Hexagon, the most perfect shape in the universe.
The slope of constant zero function is also always equal to its value, namely, zero. So it is false that exp is the only function with this property.
Can someone fix the audio in this video? Can barely hear a thing with max on.
I missed the link to prime numbers. Maybe because its hidden?
3:26 p[n]%floor(sqrt(n)) has the same kind of silhouette
nice 😊
i don't unsterstood the link with prime number
I would propose to reupload the video with much louder sound and delete this one. Anyway, the explanation is very clear and interesting.
Very Interesting. However, it seems that by 1:20 you leave prime numbers behind. I am not seeing the "link".
wait... how did this link back to prime numbers?
Actually, it didn't really. Or only in a _very_ vague way... we get the average number of divisors, and the prime numbers are the special case with precisely two divisors.
tahnks for the video. Your audio is mixed way too quiet, though.
I'm sure what's being said is very interesting. I have to assume since I can't actually hear anything.
e^x is like the identity element of the derivative operation.
I just derived* e^x an infinite number of times in 0 seconds.
*or maybe I should say "derivated"
That is not how identity elements work. When we talk about identity elements, we are talking about binary operators, not unary operators. It is more accurate to say that the exponential function is the fixed point of the derivative operator, up to a constant multiple.
is anyone else having trouble hearing the audio?
that's a nice video, but the volume is low even at maximum
woah
Before playing i guessed it grows like O(ln(n)) 😂 ln is everywhere in analytic number theory...
you should coprimes next!
Volume please
how I would approach is:
instead of counting the amount of factors a specific number has up to n, count the amount of times a specific number would be a factor of a number up to n, so for two every other number would have it as a factor and you would add n/2, for 3 every third number would have it as a factor etc, then the sum of all the factors up to n would be n/2 + n/3 + n/4… n/n, which will approach n ln n, which over n equals n
that’s also where the Euler Macheroni constant comes in, from the transition from the harmonic series to the natural logarithm (the difference between the natural logarithm of x and the sum of the harmonic series up to x approaches this fabled Euler Macheroni constant)
Man e shows up so much
oh god
I can hardly hear anything.
Approximating erroneous assumptions.
Learn to set the recording volume control, so I don't have to crank my volume up to 110% to hear your whispers.
Prime number constant......ㅎ
Uau
0:42 "no other number with this feature"
what about 0? :p /joke
Great video though :) the connection to y=1/x and lattice points was surprisingly simple and beautiful.
redo this plz
ㅎ
Stupid background music almost had me quit. Only stayed because I badly wanted to see where this was going.