The hidden link between Prime Numbers and Euler's Number
Vložit
- čas přidán 11. 12. 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.
Great video, the sound should be a little louder as the volume of this video is low compared to other videos :)
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.
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.
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.
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 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!
7:30
Appreciation to you. This should be one of the most suggested videos
amazing. you choose the best topics, and explain them beautifully.
Just ran into this video. Amazed by the thought! Thanks!
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.
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!
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 :)
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.
Gorgeous video. Bravo!
Wow, connecting the sum of divisors to the integral of the reciprocal is very intuitive but I never thought about it that way.
This is so beautiful, thank you so much for this.