Any fun integrals today?

well not really I have been struggling with a ramsey theory thing for the last few days would be nice to do integrals instead

What's Ramsey Theory?

(Pardon my ignorance)

ssorry was brushing my teeth

np

the thing I want to understand is basically if you are coloring in a cube there'l always going ot be an arithmetic progression inside it

if we make the dimension of the cube high enough with respect to number of colors and width in the cube

@cassandra0 Progression?

like

a straight line which all the same color

What do you mean by colouring a cube? The faces?

sorry I explained it really badly, but this will help me be much clearer in my own head if I get this right

Ok. Glad to help, hahaha!

so imagine a 5 dimensional cube divided into n*n*n*n*n subcubes

and we color each one in one of 3 colors say

Ok

then there should be a straight line or diagonal line or something, inside it, which is all the same color

A diagonal of coloured cubes, I presume?

@aDangerousIdea I can't make myself study though, I mostly dont even care to or enjoy doing it so I do something else to take my mind off doing poorly in school

@Jordan What do you like to do?

@Argon I am exceptionally lazy, but mostly video games, TV, reading and of course wasting a lot of time on reddit

@Jordan Haha reddit

Yeah, it is an easy way to spend time doing nothing

> In other words, the higher-dimensional, multi-player, n-in-a-row generalization of game of tic-tac-toe cannot end in a draw, no matter how large n is, no matter how many people c are playing, and no matter which player plays each turn, provided only that it is played on a board of sufficiently high dimension H.

that's actually quite a good way to say it

any number of players, any width of board... there is a dimension high enough that the game will end in someone winning

Cool! I sorta understand it :)

simply because any board filled in with that many colors will have a monochromatic line in it

the proof is a really hard to follow double induction

Hmmm... Could you have a factional dimension?

if something has fractional dimension it's a fractal isn't it

No idea. Just throwing it out there!

@cassandra0 I don't think so

At least, not necesarilly.

Hi @Peter!

Maybe?

@PeterTamaroff, why not

Cantor's set has "Hausdorff" dimension $$\log 2/\log 3$$ IIRC

that's a fractal!

hello there!

Hi

@PeterTamaroff hola, como va todo?

@WillHunting

I like the calculations about holomorphic functions which have number theory consequences

have you seen parametrizations $\delta$, $\gamma$ of the unit circle in $\Bbb R^2$ (or part of it) so that $\delta\circ\gamma^{-1}$ is not of class $C^\infty$?

@WillHunting Yep

How about you?

@WillHunting Why?

@WillHunting I see. You are a banana.

@WillHunting Why apple?

@WillHunting And B for [CENSORED]. QED

@WillHunting J for Jambul

@WillHunting Hahahaha!

@WillHunting Okayz

@Argon, do you know any nice number theory?

@cassandra0 Not really. My math is sadly quite limited.

Prime number theorem is cool!

prime number theorem is really hard, but it's related to the zeta function you can do a lot of analysis of it

@WillHunting Theorem, sorry

something like the zeta(1 + it) I think has no zeros

$$\pi(x) \sim \frac{x}{\log x}$$

Which is awesome

I actually have to learn the prime number theorem

I don't really understand it

It's really deep

RH is just a little side remark that Riemann made while proving PNT.

@WillHunting Your favourite hypothesis!

XD

there's a combinatorics bit due to Chebychev where you prove that c_1 log(x)/x < pi(x) < c_2 log(x)/x

that's quite simple actually I understand that

Then, I guess, you must show that $c_1, c_2 \to 1$

oh no these are fixed constants

1 and 1?

haha

:)

I didn't actually calculate them I should do that

Bleh, I need to learn more math

what does $\sim$ mean?

Asymptotic

$f(n) \sim g(n)$ means $f(n)/g(n) \to 1$

limit as n to infty

$$f(x) \sim g(x) \iff \lim_{x \to \infty} \frac{f(x)}{g(x)} \to 1$$

thanks

I'm just trying to understand why $c_1 \log(x)/x < \pi(x) < c_2 \log(x)/x$ doesn't give that

I mean suppose $c_1$ is like $10^{-32}$ and $c_2$ is $2^7$

I will calculate them at some point from the proofs I've got but that'll do for now

Divide all by $\frac{\log x}{x}$

Then $$c_1 < \frac{x\pi(x)}{\log x}< c_2$$

Right?

I see

And as $x\to\infty$, $c_1 < 1 < c_2$ by PNT

so what PNT is telling us is that $\pi(x)$ actually converges TOWARDS $y = \frac{\log(x)}{x}$. it's not just $O(\frac{\log(x)}{x})$

is that right?

I think it's right..

Hmmm....

like maybe primes 4k+1 converge towards the line $y = \frac{1}{2}\frac{\log(x)}{x}$

seems that

$\pi(x)$ converges towards $y = \frac{\log(x)}{x} + c$

No need for $c$, as it becomes insignificant, no?

it would be in $\pi(x)/y(x)$

become insignificant there

@leo Ah! Me perdi tu mensaje.

but I think $\pi(x)/y(x) \to 1$ implies only than $\pi(x)$ tends towards $y(x)$ plus some constant

@cassandra0 What are you discussing?

@PeterTamaroff, jus trying to understand the meaning of the result of prime number theorem

Consider the following $$\lim_{x\to\infty}\sum_{n=1}^x \frac{1}{x} - \log x = \gamma $$

@cassandra0 " The prime number theorem gives a general description of how the primes are distributed amongst the positive integers."

But still $$\sum_{n=1}^x \frac{1}{x} \sim \log x$$

@Argon That is awfull notation!

Hehehehehe

oh my gosh this is so confusing

@PeterTamaroff Oh noes! :)

why bloody way up does the fraction go

?

@Argon, what you've said I understand

Ok

@Argon Not necesarily. We can say that when $x\to a$ with $a\in \Bbb R$

For example, $\sin x \sim x$ when $x\to 0$

@PeterTamaroff Not necessarily what?

Oh, yes

Or more nicely $\sin x= x+o(x)$ when $x\ to 0$

You are right, the limit must be given somewhere

see n/log(n) seems to be a constant away ?

@cassandra0 If you really want to understand the prime number theorem, you should study some analytic number theory first

@cassandra0 There exists no constant $C$ such that $\pi(x)+C = \frac{x}{\log x}$

@Argon I think that is courtesy of Chebyshev

@Argon, I know but isn't there a constant $C$ such that $\pi(x) + C$ converges towards $\frac{x}{\log x}$? In the sense that $|(\pi(x) + C) - \frac{x}{\log x}| \to 0$?

I just don't really understadn what $\pi(x) \sim \frac{x}{\log(x)}$ tells us

Do you know little o?

yes

$f\sim g \implies f(x) = g(x)+o(g(x))$

I see

so if we have $\varepsilon > 0$ there's some $x$ such that $|f(x)-g(x)| < \varepsilon |g(x)|$

but $g(x)$ grows so the bound could be as large as anything

Yes

so why is PNT actually stronger than Chebychevs result?

Not sure. I really don't know enough number theory to give a good answer.

$$c_1 \log(x)/x < \pi(x) < c_2 \log(x)/x$$

this is chebychevs result

@PeterTamaroff, any views on this

@cassandra0 Yes.

Yay!

Pedro to the rescue.

:)

Hi @Will !

I actually asked about it

Jambul?

HAHAHAHA!

this is very useful to me but it's not the same result of chebychev

@WillHunting We both have quadrilaterals as our avatars.

@WillHunting My use of ellipses is contagious!

@amWhy Are you Charlie's replacement?

Hahahahaha!

@PeterTamaroff How so?

hi @amWhy

@amWhy Well...

do you do number theory?

@cassandra0 Hey!

Charlie is in this room!

That is completely inappropriate!

@cassandra0 hi cassandra!

sorry I just wanted to know

@PeterTamaroff Geez

:)

@cassandra0 I'm just punning around.

I have to prove some stuff.

Let's get serious.

@Peter I like your avatar

I don't what happened to Charlie?

I'm clearly missing something...

@Argon It is nice, right? Plushies!"

@PeterTamaroff Haha! They are awesome

Hi @amWhy, by the way!

Hi, @Argon, how goes it?

@amWhy Good, good

You?

@Argon Pretty good =)

@amWhy How do you define a dense subset of $\Bbb R$?

Good @Will Hunting

@WillHunting Why not?

analytic number theory

I'm thinking: $A$ is dense in $\Bbb R$ if for every $x\in\Bbb R$ there exists $\{a_n\}\subset A$ such that $\lim a_n=x$

@WillHunting No topological considerations, man,

But yes, then I'm right.

and $a_n\neq x$ right?

@PeterTamaroff pretty much topologically: "E" dense subset of \Bbb R if every point of \Bbb R is a limit point of E, or a point of E, or both.

How would I go about finding the general (non-recursive) forumula for $$t_n = 2 t_{n-1} + (-1)^n$$

@WillHunting That would allow isolated points.

@WillHunting You mean XXX

@Argon Dude.

@PeterTamaroff Oui?

@Argon Analyze odd and even parts.

Profit.

@amWhy Right

@PeterTamaroff Parts?

@Argon subsequences

@WillHunting Limit points vs. isolated points.

Its right there, black and white,.

@PeterTamaroff Can you explain what you mean?

hi