I'd suggest trying some low-degree examples e.g. $f(n)=an+b$

@Daminark yes

Isn't defined it defined as $$\zeta(s)=\sum \frac{1}{n^{s}}$$

Sums need limits...

That's right, so the idea is that in complex analysis you've got this nifty thing called analytic continuation

@Daminark what's analytic continuation if I may ask ?

Basically, the idea is this

Let's take an open set $U\subset \mathbb{C}$

ok

And let $f:U\to\mathbb{C}$ be analytic

Now, take some larger open set $V \supset U$, and some function $F$ which is analytic on it

so the function is on our open set and analytic, what happens from there ?

Actually, consider two such functions, $F$ and $G$

Where we have that for $z\in U$, $f(z) = G(z) = F(z)$

Then you have that for all $z\in V$, $G(z) = F(z)$

So that's the idea, you extend the domain of an analytic function

ahhhh ok so the layman way of putting it is we extend the amount of imputs we have to analytic function

Yup, and in doing so we maintain analyticity

Now, of course you can't just do it free willy nilly, you can only sometimes do it

To properly talk about when you're allowed to do so you use this thing called the monodromy theorem, which is about sheaves and germs and all that

But it is pretty nifty when it can be done

@Daminark what's the connection to ANT

You can also define $L(s)=\sum\frac{\chi(n)}{n^s}$ for any function $\chi$

Turns out you can analytically continue the Riemann zeta function, which in its series form is only defined when $\Re(s) > 1$, to a function which only has a pole at $s=1$ and should otherwise by analytic

@Daminark and that allows for one to introduce the notions of convergence or divergence of the zeta function

and usually $\chi$ is taken to be multiplicative, which means that $\chi(ab)=\chi(a)\chi(b)$ when $a$ and $b$ are relatively prime (so it's determined by its values on prime powers)

Now, the connection comes when you see that $\zeta(s) = \prod_{\text{primes}} \frac{1}{1-p^{-s}}$

Oh yeah all that Dirichlet character stuff

It's neat

@Semiclassical Would this still hold for the more general $f:\Bbb N\times\Bbb N\to\Bbb Q,g:\Bbb N\to\Bbb Q,g(n)=\sum\limits_{k=1}^nf(n,k)$?

Yeah so the TL;DR is that estimating $\zeta$ tells us stuff about primes from that formula

and the formula is really easy to prove, actually; expand out the geometric series

But yeah that's like, one example in which it comes about. The Riemann hypothesis is a statement that the non-trivial zeroes of the zeta function (the trivial ones being negative even integers) all have real part 1/2

ahh ok, at @AkivaWeinberger @Daminark it seems the way to properly to talk about ANT is via a constrcutive appoarch

@Daminark that's exetermly compelling

@Daminark what's another example of what ANT is ?

If that's true, it gives a nice bound on the difference between the prime counter and the logarithmic integral

Look up the prime number theorem

I mean, I don't know much analytic number theory, I don't even really know much complex analysis

Also I don't understand entirely what "constructive approach" means

@Daminark dang :>(

@Daminark explatnation by example

Well, I mean there's a simple way to describe it

Which is, analytic number theory is what happens when you use tools of analysis (especially complex analysis) to derive results in number theory

ahhh ok @Daminark

Now, that was the one example I know of offhand to demonstrate why that's a thing, since a priori you might not necessarily have reason to believe that analysis should come up in number theory whatsoever

true

@Daminark it's not really that the tools of analysis "show up" in number theory, but that the tools of analysis end up being useful for number theory. Nothing really "shows up" in anything else so much as perspectives can be elucidating.

And I mean, it's not even surprising that analysis "should be used" sometimes given that the kind of problems analytic number theory tries to solve are kind of analytic type questions to begin with

Hey guys

@AkivaWeinberger yeah nevermind. Forget about it. I wasn't thinking it through.

that is true @EricSilva

anyways

hey ya'll, I am not joking when I say I have proven the inductive step of a collatz conjecture proof aside from the cases of 12n + 3 and 12n + 7. Any.... thoughts?

also, didn't someone once prove the conjecture for some cases in mod 32 or something? I couldn't find it on wikipedia.

My point was more, that number theory had analytic type questions

Is not immediately obvious

distribution type questions are pretty natural

"are there infinitely many primes" is basically an analytic type questoin

It's easy enough that it doesn't need analytical tools bc it's like the most basic distribution question

but the analytic approaches are honestly cooler than the 2000 year old one (even though that's an A+ proof imo)

@EricSilva hands down the best proof is ALWAYS whatever proof is most concise and easy to read for the audience you desire to share it with.

of course, if only one proof exists then that is what you use.

…What was the context

^^precisely

but even if the context of mathematical knowledge is the same, the best proof is the most concise in a writing sense.

I don't buy the notion that you can well-order proofs as such

@Typhon the "best" proof depends what you need

"Unless you already own a sledgehammer and want to be a bada**, don't forge one to break a cracker."

the professor once said that in the proofing class I had.

in other words, don't overdo it.

@Typhon The "best" proof depends on context, I mean. It depends on what you're using it for

once again

I mean, sometimes if the sledgehammer is useful in its own right...

ugh

And you also have a cracker, like, multitasking

let us assume your only goal is to break the cracker

with theorems a,b, and c

Which is almost never the case but alright

don't prove theorem z to break the cracker unless you need to.

I think this is a statement that's so absolute and broad that it's like definitely false

or rather

don't include the proof of z within the breaking of the cracker

just use a citation

Okay, if your only goal is to prove the theorem and there's no other context, then yeah the most elegant one with the fewest big ideas wins

harder proofs or proofs with bigger tools can be the "right one" for your circumstance depending on things like expediency or introducing new ways of thinking into old fields

I wouldn't even say that necessarily is true

of course

@Typhon, did you star that? You do realize I'm not trying to defend you, right?

I agreed with it

@AkivaWeinberger I think I might be able to prove the collatz conjecture.

the most elegant proof of the prime number theorem ive seen with the fewest big tools is also the hardest one i know

OK, it just feels odd

it's also the least useful one in terms of where you can go with it

@Typhon Go on…

@EricSilva Let's assume that to use those tools, you'd have to prove them.

It tells you something about the problem you're working on, and it tells you how those big ideas relate to what you're doing

all of them

@Typhon sometimes it is genuinely easier to develop the theory.

@AkivaWeinberger Basically I had a crazy thought.

think of it like this

you have a tree

with branches a, b, and c

@Daminark basically what I meant when I said sometimes it injects new ways of thinking into old things

@Typhon the issue here is that you're assuming a basically nonexistent situation. You're never just like, I want to prove this theorem and care about nothing else at all

if b is connected through c to some point 1

@Daminark alright. I get it. I was merely trying to point out that proofs shouldn't be 20 pages long for the sake of length.

(extreme example)

Philosophizing is best done with a concrete example first, and then abstractions

but no one does that

So like sure, if you develop contrived motives you can manufacture all these hypothetical imperatives, but that's not the point. Like I personally prefer shorter proofs anyway, and often... "talk-y" proofs

OK

except federer

Wait what were you referring to when you said that no one does "that"?

no one makes a proof long just for it to be long

@AkivaWeinberger My issue here is more that in many such cases you can't nicely generalize from too many specific examples

and besides long proofs are frequently easier than short ones

@AkivaWeinberger Basically by induction I can say 11 fulfills the collatz conjecture because if you go backwards 11 could come from 22 which could come from 7. However by induction 7 goes to 1. Therefore, 11 MUST go to 1, otherwise 7 wouldn't. I'm currently working in the cases of 0 through 11 mod 12. I can prove all except 7 mod 12 and 3 mod 12 without having to use the proof of the other cases within the inductive step. Basically, I merged the idea of going backwards with going forwards.

By "many such cases", I mean trying to talk about things being "better" in math, and likely most other subjects

and then I traverse the 'tree' to solve different cases. The trick is dealing with straggler cases.

for some reason I thought I read somewhere that (ironically) those cases were proved by someone else.

so then merging the two would complete the proof of the collatz conjecture

It sounds like you're trying to prove that every number appears in a Collatz sequence generated by a smaller number

That's a testable claim. Write a program to see if it has exceptions.

Because once you zoom in on that specific example, you've added new conditions/ends such that a hypothetical imperative does begin to surface, but it was too situation-dependent. As we did here, if we are to assume that someone has no desire but to learn theorem X and that this person values efficiency, yeah sure the shortest proof, by whatever the relevant metric is, accomplishes those goals best

Theorems aren't the end goal

Just thought I'd reference that quote again

But try extending that to a universal statement, you realize that oh look, those are oftentimes very much not the ends of everyone. Oh well. In part for this reason I tend to be skeptical of many such normative claims

Gian-Carlo Rota is eminently quotable

What's funny is that the goal is to have fun

who's having fun?

that's not allowed

To some degree

As one of the reviewers of one of my books once told me, mathematics is no place for fun or humor.

The goal is to have fun, I never said you needed to have fun

Sad people

I'd be a bit cautious about reducing too much to a kind of hedonistic view, it's definitely there, but I might generalize slightly to satisfaction

I was in shock over such a comment.

@AkivaWeinberger no.... it is a combination of that and showing smaller numbers occur further in the sequence. For instance, 12 goes to 1 by induction because 12 is proceeded by 6.

@TedShifrin Was this in response to a specific passage in your book?

It was in response to my injecting occasional humor. God only knows what that reviewer would have thought of watching my classes ...

I like what Thurston's essay says on how we make progress in math

our personal goals in doing it can be whatever though

I believe the correct response would be "Who hurt you" @TedShifrin

@EricSilva What does it say?

read the essay: math.toronto.edu/mccann/199/thurston.pdf

Sorta like one of my old colleagues (and department head for one term) who said he didn't see the need to collect and grade proofs in upper-level math courses. After all, he said, no one had ever graded his work and look how he turned out. (No comment.)

it's good and short

@Typhon Oh, I see. So, each number either is contained in the Collatz sequence of a smaller number, or contains a smaller number in its own Collatz sequence. And you want to prove it by going through each equivalence class mod 12.

@AkivaWeinberger exactly

i have yet to prove the case for 12n + 3 and 12n + 7.... I think the latter is about to be proved.

Oh, Eric, that's the essay that led Zev to post a question on MSE asking for an explanation of one of the definitions of derivative. It's one of my most highly-voted answers.

Hm. Consider the statement $C_n$ to mean, "Each number either is contained in the Collatz sequence of a smaller number, or contains a smaller number in the first $n$ steps of its own Collatz sequence" @Typhon

Zev Chonoles?

So, like a shallower version of the thing

Yup.

oh he's cool

Smart dude. I was proud of my response. :)

@TedShifrin Why me

he sold me a shirt

Oh isn't he one of Peter's people?

and has good notes

Oh, good point, DogAteMy. Sorry.

Then we can ask if any $C_n$ is true

yeah @Daminark

Yeah the one who wrote notes for Marianna's class, that's dope

I thought he went into alg geo, but I haven't kept up.

Answer in question: math.stackexchange.com/questions/449740/…

Oh, that sounds all oxymoronical, doesn't it, "answer in question"

Equivariant algebraic topology, as it turns out

@Ted I think he likes alg geo and number theory

Thanks, Demonark.

he says something on his web page iirc

Yeah Eric's right

Question in answer? DogAteMy

"tongue firmly implanted in cheek" is an expression i might steal

Turns out Thurston had a small (sign?) error in his thing, if I remember right.

Make sure you send me royalties, Eric :P

it's worth it, that's quality right there

:)

@Typhon The reason I defined that is that I bet your proof strategy could only work if one of the $C_n$ were true (and with a small $n$, most likely)

Perhaps a computer could calculate what $n$ each number would have. If it's bounded, that's great news

@Ted this is a great answer

Thanks, Eric. It means a lot that you say that.

@EricSilva I Google "contranegative" and it leads me back to that paper

or rather

someone else who was reading the paper and asked on Stack Exchange what it meant

Contranegative? :)

i have no idea what contranegative could mean

It's in the introduction to the essay

I guess he just meant negative, but, well, weird

isn't he making a joke

No, I get what he meant. He meant denying the negative. Look what's become of American politics. It's all contranegative, nothing positive.

He was of course punning on contrapositive.

yeah that's what i assumed

I miss the days of really interesting questions on MSE ...

@Thurston re sound effects: Can relate

i wish i couldve met that dude, i've heard his intuition was insane

Actually Marianna told us this one funny story in office hours

Yeah, amazing topologist/geometer. Not always the clearest lecturer, as I recall, but still astonishing.

We were talking about something and at one point I was just like "Yeah so you do this and get it goes joop, so you're done"

Hi chat

Benson told me a story about how he told him a vague idea of what he wanted to do for his thesis and Thurston just blurted something out instantly that it took Benson the rest of grad school to figure out

Rehi @Alessandro. (Oh yeah, I was thinking finite-dimensional, sorry.)

Oh, I had such experiences with Chern and Griffiths repeatedly, Eric.

that's crazy

And Marianna mentioned that a few people who didn't know English had come up with something which they named "Pu-pu", and there was a reason for this I don't remember, but it... was very unfortunate

And oh snap

@Alessandro yo

Ok, I agree now. I still didn't know that all functionals are obtained as scalar product with a vector but that makes sense

I was expecting Marianna's story to end at Thurston, Demonark.

yeah it kind of felt like a non-sequitur

Oh I mean, it was just a tangent I was reminded of due to Thurston's comments about sound effects

ohhh

very obscure

In my case, I use humor a lot ... most would say bad humor. Although I have had a few students who engaged in pun contests with me in class and sometimes it was just ridiculously funny.

@Alessandro this fact can be pretty amazingly useful

Did those result in multiple smacks and/or eye-rolls

No, smacks and eye-rolls are an MSE chat thing. Brought about by y'all.

(A compilation)

I wonder what the high school kidlets will make of my "demeanor."

It shocks me to now, our humor is pretty high quality!

eight pages of smacks, DogAteMy? and how many of multiple-eye rolls?

Someone who doesn't know me might try to get me thrown in jail.

Demonark, "to now"?

Better way to word it along similar lines might be: "I'm shocked to this day at the smacking!"

what are you saying

Your humor is usually worse than Balarka's, Demonark. That's setting a very low bar.

Like, I wasn't just surprised at the beginning or something, even now I'm taken aback...

You're more shocked by my smacking than Hippa was by my students' polar-coordinate forgetfulness?

ok, i couldnt decipher what you were saying because the phrasing seemed awkward to me

incoming physical intuition overload but if you're doing a headstand isn't that going to be like, a really high bar? The height of the sky even?

That's cuz it was, Eric.

yeaah

@TedShifrin Well, maybe not that shocked

:P

@Daminark the bar isn't high in that case, the sky is just really low

It appears Ocelo has reappeared with a minor name alteration? any personality alteration?

@TedShifrin I dunno if we can determine that on basis of the name change

chuckles

You probably didn't know him earlier. Balarka and Mike put him on permanent ignore, I think.

Yeah I don't know him, I was just making a determinant pun on the word "minor". But wow

Someone in here was talking I have on permanent ignore.

Oh yeah, the former Chris'ssis was talking to robjohn.

There's like three of those people and they all have indistinguishable personalities.

makes noises at PVAL's general direction

DogAteMy is going to grab your dinner, PVAL.

I what now

That's what the noises sounded like.

Who are you talking to Ted?

:)\

shrugs

So say someone asks a question on main and their question can be interpreted in multiple ways. What's like the correct response to that? I sort of feel like its the questioners responsibility to ensure questions are understandable, and while in real life I wouldn't mind having a conversation in order to help them phrase their question better, I feel like the format of this site is terribly suited for that purpose.

Take a circle on the complex plane that has the origin on it