You can write f(x) as an infinite sum with Heaviside functions, and then integrate by parts normally recognizing that the derivative of the Heaviside function is the delta function.
Just one little bit to pick: Abel summation is named for Neils Henrik Abel, a Norwegian, whose last name is pronounced AH-bel, not like the English word "able."
I wish I could understand all this stuff as a type of muscle memory. But when I try to take it in by one ear - just like Homer Simpson - some of the old stuff drops out the oither ear - : shrug :
Fun fact: the limit of 1/n * (n!)^(1/n) is instead 1/e, so by replacing the primorial with the factorial and then dividing the result by n we get the exact reciprocal as limit. Not sure what that means though ;-)
But whatever it means, we can combine both limits to find another one: If we define n? as the "anti-primorial", i.e. the product of all natural numbers 1/(e^2)
This implies the primorials have something akin to Stirling's approximation:
nth_root(n#) ~ e as n -> ∞
so n# is roughly approximated by e^n
crazy
Yes but it’s not a very good approximation unlike the Stirling approximation.
The fact that this limit is finite, swapping n# for n! turns into an infinite limit and that the number of primes is infinite. Shocking
Thanks
Numerically, I get something slowly creeping towards "e".
You can write f(x) as an infinite sum with Heaviside functions, and then integrate by parts normally recognizing that the derivative of the Heaviside function is the delta function.
Just one little bit to pick: Abel summation is named for Neils Henrik Abel, a Norwegian, whose last name is pronounced AH-bel, not like the English word "able."
That's a Lie.
11:46 Yikes
At least it was a good place... 😉
... and it is an excellent place to start
amazing derivation!
What a cute limit. Love it ! 👍❤
I think it's more like an alternative statement of the prime number theorem.
That's impressive!
How does he always know the good place to stop? 😮
He is God here. He declares the endpoint.
Do it instantly with Cauchy-d'Alembert
That's some interesting math
What a COOL RESULT !!!!!
I wish I could understand all this stuff as a type of muscle memory.
But when I try to take it in by one ear - just like Homer Simpson - some of the old stuff drops out the oither ear - : shrug :
I weigh seven/eight/nine/ten pound! 😂
That's a good place indeed.
Fun fact: the limit of 1/n * (n!)^(1/n) is instead 1/e, so by replacing the primorial with the factorial and then dividing the result by n we get the exact reciprocal as limit. Not sure what that means though ;-)
But whatever it means, we can combine both limits to find another one: If we define n? as the "anti-primorial", i.e. the product of all natural numbers 1/(e^2)
I guess for n! the solution will follow a similar path… or not?
Michael did already several videos using different methods for n!, if I remember correctly.
need explanation about PNT and the homework :D
Interesting.
I would have just said n primorial. All these pounds make it sound like a money problem, or even a mass problem.
This is a pound symbol: £
That is the monetary pound. # is the weight pound symbol.
we may be living in a post-hashtag world
Noice
Supercool… as always
Guess before finishing the video: The answer is 1.
Primes get sparser indefinitely.
Well. Your wrong
@@015Fede -- Well, you're wrong, when you misspelled "you're."
en.wikipedia.org/wiki/Primorial_prime#:~:text=In%20mathematics%2C%20a%20primorial%20prime,sequence%20A057704%20in%20the%20OEIS).
Did you mean https ://en .wikipedia .org /wiki / Primorial_prime# |?|