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."
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.
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)
Numerically, I get something slowly creeping towards "e".
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
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.
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.
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.
11:46 Yikes
At least it was a good place... 😉
... and it is an excellent place to start
What a cute limit. Love it ! 👍❤
That's some interesting math
How does he always know the good place to stop? 😮
He is God here. He declares the endpoint.
That's a good place indeed.
I weigh seven/eight/nine/ten pound! 😂
amazing derivation!
Do it instantly with Cauchy-d'Alembert
I think it's more like an alternative statement of the prime number theorem.
What a COOL RESULT !!!!!
That's impressive!
Thanks
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)
need explanation about PNT and the homework :D
Interesting.
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.
This is a pound symbol: £
That is the monetary pound. # is the weight pound symbol.
we may be living in a post-hashtag world
I would have just said n primorial. All these pounds make it sound like a money problem, or even a mass problem.
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# |?|