The hidden link between Prime Numbers and Euler's Number

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.

[drippyeuler]

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.

[JM-usfr]

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.

[merrickdodge]

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!

[josephyoung]

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"

[killymxi]

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.

[stanleydodds]

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).

[joseville]

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.

[qulaeygaming]

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.

[j.vonhogen]

This is an excellent video. Please, make many more of these!
Thanks a lot!

[gustavocortico]

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?

[Vannishn]

Good video ! But don’t we have some multiple of the Euler mascenori constant as the limit of the difference ? 7:50

[MuhammadFaisal_Iqbal]

00:03 Euler's number e has a special property related to the exponential function.
01:27 Finding formula for average number of divisors of numbers 1 to n
03:10 Average number of factors of numbers shows smooth growth
04:44 Graphing divisors in the x-y plane reveals a hidden link between prime numbers and Euler's number.
06:09 Visualizing divisors using points
07:42 Using calculus, relate the number of factors to the area under a curve.
09:19 The link between prime numbers and Euler's number lies in the relationship between exponential and logarithmic functions.
10:52 Numbers from 1 to n have on average log n factors.

[PeterParker-gtxl]

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.

[omerelhagahmed]

Appreciation to you. This should be one of the most suggested videos

[antoniorose]

Just ran into this video. Amazed by the thought! Thanks!

[Darkstar]

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

[rotemperi-glass]

amazing. you choose the best topics, and explain them beautifully.

[vivada]

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?

[Vito-jrwl]

Great Video and pleasant voice and background music!

[elephantdinosaur]

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 :)