ITT Veeky Forums solves a Millennium Prize Problem

I actually did some work on the RH a little while back:
files.catbox.moe/1c46nx.pdf
Sorry if it's a bit hard to make sense of. The symbols are basically the ones Mathematica uses.

Why does he look ugly in this pic? I thought he was handsome

The fuck are you talking about? He looks as sexy as ever.

...

>If some random Russian dude could figure out the Poincare conjecture

I did a little more work on it:
files.catbox.moe/6atcqt.pdf
Li's criterion says that proving [math]\lambda_n > 0[/math] for all [math] n \in \mathbb{N} [/math] is equivalent to proving the Riemann hypothesis.

>set up product integral function [math] f(a)=\displaystyle \prod_{-\infty}^{\infty}\zeta(a+xi)^{dx} [/math]
>for some fixed a, if the zeta function has a zero along the line a+xi, then f(a) is going to be 0
>use real induction and prove that f(a) is nonzero and divergent on the set (0,1) \ {1/2}
>you have just proven the riemann hypothesis

[math] \displaystyle \prod_{-\infty}^{\infty}\zeta(a+xi)^{dx} = \exp(\int_{\mathbb{R}}\ln(\zeta(a+xi))dx [/math]
[math] \displaystyle \zeta(x) = \prod_{p\text{ prime}} \frac{1}{1-p^{-x}} [/math]
[math] \displaystyle \ln(\prod_{p\text{ prime}} \frac{1}{1-p^{-x}}) = \sum_{p \text{ prime}}\ln(\frac{1}{1-p^{-x}}) [/math]
[math] \displaystyle \sum_{p \text{ prime}}\ln(\frac{1}{1-p^{-x}}) = \sum_{p \text{ prime}}(\ln(1)-\ln(1-p^{-x})) = -\sum_{p \text{ prime}}\ln(1-p^{-x})[/math]
[math] \displaystyle \exp(\int_{\mathbb{R}}\ln(\zeta(a+xi))dx = \exp(-\sum_{p \text{ prime}}\int_{\mathbb{R}}\ln(1-p^{-a-xi)})dx)[/math]
[math] \displaystyle \exp(-\sum_{p \text{ prime}}\int_{\mathbb{R}}\ln(1-p^{-a-xi)})dx) = \prod_{p \text{ prime}}\exp(-\int_{\mathbb{R}}\ln(1-p^{-a-xi)})dx) [/math]

it gets messier

[math] \displaystyle \int_{\mathbb{R}}\ln(1-p^{-a-xi)})dx = \frac{i\text{Li}_2(p^{a+bi})}{\ln(p)}+b\ln(1-p^{-a-bi})-b\ln(1-p^{a+bi})+\frac{1}{2}ib^2\ln(p)+C [/math]

continued in next post

Although I have not examined its mathematical content, I wish you to know that I saw your [math] \LaTeX [/math] and found it unusually beatiful, at a glance.

Two issues which immediately pop out are the exponential "dx" in the beginning, and small parentheses throughout, which do not match the large operators.

Your post brings to mind two things:

1) What /is/ THE Riemann Zeta function, over the complexes, once all the brouhaha has been accounted for? A single, singular formula, which isn't just the early real motivation. This is something I've never actually checked.

2) Waht is the "Li" function? I can readily search this but I'm inviting you to add context, is the point.

you messed up your latex pretty bad
you forgot a parenthesis on line 5 and changed an x to a b on the last line
also, i'm hoping you just copy/pasted that integral over R, because that's a definite integral with a constnat of integration

[math] Li [/math] is usually the logarithmic integral function.

stare zero and let halves come

>wants to solve perhaps THE greatest mathematical problem of the 20th century
>can't even into LaTeX right
Wonderful. Just, friggin, wonderful.

Here's a few more hours worth of work on it:
files.catbox.moe/ri9i0v.pdf
Prove the [math]\lambda_n[/math] sequence is monotonically increasing, and by reference to the fact that [math]\lambda_1 > 0[/math], the result follows by ordinary induction.

Bump :^)

Instead of just mindlessly bumping, how about you help?

If we can determine some sort of closed-form answer for the incomplete Bell polynomial at the bottom, the we can remove the last derivative operator. Then we just take [math] \lim_{z \to 1} \lambda_n [/math] and we'll have a simple genral form for the terms in the series, which should make it much easier to analyze, seeing as doing the same for the pre-proceesed cases of [math]n = 1[/math] and [math]n = 2[/math] appear to result in a series expressed in terms of the various Stieltjes gammas.

>real induction
>no discussion of the choice of logarithm
>no discussion of convergence whatsoever
>no words

You do know that proofs usually have words in them right ? At least explain what you're doing if you are expecting anyone to even glance at it

It's just scratch work, f a m. Though I agree his work would be more intelligible if he explained in words.

Still, he could at least explain why he is making these computations and why he thinks they are relevant. He seems to expect people to help. The least one can expect is to know what is happening. It's like asking someone to read disgusting, non-commented code

I explained why they're relevant in and and what I'm doing is attempting to put it into a form for which a growth rate can be determined.