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The Fundamental Theorem of Probabilistic Number Theory - Shared screen with speaker view
espen slettnes
09:07
Handout: https://espen.slett.net/BMC_lectures/Probabilistic_Number_Theory/2020-06-16.pdf
Jake Robb
12:12
yes
Sheldon Tan
12:15
yes
Derek Liu
12:16
yees
Justin Kang
12:18
yes
Derek Liu
13:48
and it doesn't go to 0 right?
Derek Liu
14:19
oh
Derek Liu
14:42
x^1.99998
Sheldon Tan
14:42
5x^2
Leo Xu
14:43
quadratics
Sebastian Weinberger
14:46
ln(x)
Nathan Barnes
14:49
x
Derek Liu
14:51
xlnx
Derek Liu
14:56
x ln x ln ln x
Nathan Barnes
15:00
1
Leo Xu
15:01
constants
Derek Liu
15:02
1/x
Jake Robb
15:02
0
Sebastian Weinberger
15:04
ln(x)/(x^2)
Derek Liu
15:04
1/ln x
Derek Liu
15:09
1/(ln(ln(ln x)))
Sheldon Tan
15:10
1/x^7987
Derek Liu
15:13
9
Derek Liu
15:47
x
Eddy Li
15:48
1/x
Sebastian Weinberger
15:49
ln(x)
Sheldon Tan
15:49
1/x
Andrew Sylvester
15:49
1
Derek Liu
15:53
1
Sebastian Weinberger
15:55
x^2/ln(x)
Derek Liu
15:58
does 0 count?
Sheldon Tan
15:58
x^1.8889898989
Derek Liu
16:03
x^1.999999999999999999999999
Derek Liu
16:07
x^(2/x)
Eddy Li
16:25
x^(-1.24567)
Derek Liu
16:41
wait x^(2-1/x) wouldn't work, right?
Sebastian Weinberger
16:55
It does not, I think
Eddy Li
17:14
I don't really understand the big O function and the theta function
Sebastian Weinberger
17:23
Because lim(x->inf)(x^(1/x)) is finite and nonzero
Sheldon Tan
17:32
the big o function is when the limit is not infinity
Sheldon Tan
17:37
of g/f
Derek Liu
18:57
they approach 1?
Sheldon Tan
19:07
so they get arbitrarily close to 1
Leo Xu
19:12
wouldn’t x^2/x^1.9999999 eventually approach infinity
Derek Liu
19:13
uh should this have absolute values?
Derek Liu
19:28
o(f)
Derek Liu
19:30
?
Sheldon Tan
19:52
what?
Derek Liu
19:58
f+o(f)/f approaches 1 always
Derek Liu
20:05
and f-g/f approaches 0
Sheldon Tan
20:17
o
Derek Liu
20:55
1/ln n
Andrew Sylvester
21:19
0
Eddy Li
21:20
0
Jake Robb
21:21
0
Owen Xu
21:21
0
Nathan Barnes
21:26
1
Andrew Sylvester
21:26
1
Owen Xu
21:26
0
Jake Robb
21:26
1
Sebastian Weinberger
21:27
1
Leo Xu
21:27
1
Eddy Li
21:29
1
Sebastian Weinberger
21:30
3
Leo Xu
21:31
3
Nathan Barnes
21:31
3
Eddy Li
21:31
3
Andrew Sylvester
21:31
3
Jake Robb
21:31
3
Sheldon Tan
21:32
3
Derek Liu
22:16
pi(x)=1/lnx
Derek Liu
22:25
wait sorry x/lnx
Seonghyun Yoon
22:28
x/lnx
Sheldon Tan
22:29
n/ln n
Sheldon Tan
22:31
x
Derek Liu
22:34
im used to hearing the density version
Jake Robb
24:33
ln2
Zihongbo Wang
24:44
ln5
Derek Liu
24:45
ln5
Sheldon Tan
24:46
ln5
Andrew Sylvester
24:46
ln5
Sebastian Weinberger
24:46
ln(5)
Jake Robb
24:47
ln5
Derek Liu
24:55
ln1
Zihongbo Wang
24:56
0
Nathan Barnes
24:57
0
Sebastian Weinberger
24:57
0
Sheldon Tan
24:57
ln1
Andrew Sylvester
24:57
0
Sheldon Tan
24:59
0
Sheldon Tan
25:32
so ln 1 is 0 isnt it?
Maxim Tarima
25:39
...
Maxim Tarima
25:43
ya
Sheldon Tan
25:46
lol
Maxim Tarima
26:01
sure
Derek Liu
26:18
plus 250 is not a power of 1
Sheldon Tan
26:25
big brain]
Derek Liu
26:52
we can consider each prime individually
Derek Liu
27:09
~log_2 x powers of 2, log_3 x powers of 3, etc.
Derek Liu
31:10
a
Derek Liu
31:15
floor(log_p x)
Seonghyun Yoon
32:31
lnx
Sheldon Tan
32:34
um i dont think that ln_p exists
Maxim Tarima
32:45
i was wondering lol
Seonghyun Yoon
32:59
ln(x)
Derek Liu
33:48
so we get ln(x^pi(x))?
Derek Liu
34:14
pi
Sebastian Weinberger
34:15
pi(x)
Derek Liu
34:16
(x)
Sebastian Weinberger
34:35
Big O?
Eddy Li
35:01
but isn't O not a function
Derek Liu
35:18
on the order of x
Derek Liu
35:22
at most
Derek Liu
35:25
the order of x
Eddy Li
35:48
so ln(x)pi(x) is big o
Sebastian Weinberger
36:16
We can also prove theta(x) by putting bounds on floor(log_p(x))?
Sheldon Tan
36:17
but o isnt a function'
Eddy Li
36:56
so how can O have a value
Sheldon Tan
37:00
yeah
Derek Liu
37:59
@Sebastian we're assuming pi(x)=O(x/lnx), not theta(x/lnx)
Derek Liu
38:01
I think
Sebastian Weinberger
38:19
Got it
Derek Liu
38:43
floor(x/d)
Sheldon Tan
39:06
man im lost
GavinAbbe
39:28
glad I'm not the only one
Nathan Barnes
39:40
O(1)
Seonghyun Yoon
39:43
O
Sheldon Tan
40:07
so is O just "less than"?
Derek Liu
40:50
O(d)?
Nathan Barnes
41:10
O(x)
Sebastian Weinberger
41:15
psi(x)?
Sheldon Tan
41:20
so big o of something means anything that satisfies that property?
Sheldon Tan
41:35
ok
Sheldon Tan
43:35
?
Sebastian Weinberger
43:40
You lost an x in an earlier step I think
Sheldon Tan
45:23
when you say the difference is O(1), can't any constant be O(1)?
Derek Liu
47:17
1/(p(p-1))
Sheldon Tan
52:05
there is 1
Sheldon Tan
52:14
and none in the denominator
Andrew Sylvester
52:16
there are none
Sheldon Tan
52:28
yip p
Nathan Barnes
53:19
because 2^2n is the sum of all k choose 2n
Derek Liu
53:23
because 2^2n is sum of that row of pascal's triangle
Derek Liu
53:29
pathcounting
Leo Xu
53:37
binomial expansion of (1+1)^2n
Derek Liu
53:37
pathcount on an n by n grid
Eddy Li
53:51
'to be or not to be': 2 ways: head or tails
Nathan Barnes
55:33
then we do π(n)-π(n/2)
Sebastian Weinberger
55:45
And add inequalities?
Derek Liu
56:49
(k-1)/2
Jake Robb
56:51
k-1/2
Sebastian Weinberger
57:57
We need to stop one step earlier because division by zero
Sebastian Weinberger
58:01
I think
Sebastian Weinberger
58:34
Oh wait I think I misread
Derek Liu
58:57
wait but all of these are of the form 2^x/x not 2^x/(x-1)
Derek Liu
59:03
shouldn't it be 4/1+8/2+16/3+...?
Sebastian Weinberger
59:09
Yeah…
Derek Liu
01:00:59
geometric series?
Derek Liu
01:01:50
te right one
Derek Liu
01:01:59
the bigger exponent
Leo Xu
01:02:00
higher power
Derek Liu
01:02:02
*much bigger
Derek Liu
01:02:36
k=O(ln 2^(2k))
Nathan Barnes
01:03:20
log(n)
Sebastian Weinberger
01:03:22
k = log_4(n)
Sebastian Weinberger
01:04:19
But still diverges!
Golden Peng
01:05:31
Thanks!
Derek Liu
01:05:31
Thank you!
Leo Xu
01:05:32
Thank you!
SheldonTan
01:05:35
Thank you!
Andrew Sylvester
01:05:37
thanks!
Sebastian Weinberger
01:05:38
Thanks!
Seonghyun Yoon
01:05:39
thank you
Jewoo Suh
01:05:39
Thank you
Zihongbo Wang
01:05:39
Thanks
Chang Yu
01:05:39
Thank you!
Owen Xu
01:05:39
thank you!
Justin Kang
01:05:44
Thank you
Eddy Li
01:05:49
Thank you
Eddy Li
01:05:50
bye