Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 14:14:19 UTC No. 16287223

Substituting [math]s=\frac{1}{2}+it[/math] and using the symmetry of the zeta function, we get:[eqn]f\left(\frac{1}{2}+it\right)=\frac{1}{2\pi i}\oint_{|s-1|=\frac{1}{2}}\frac{f(z)}{z-\frac{1}{2}-it}dz=\frac{1}{2\pi i}\oint_{|s-1|=\frac{1}{2}}\frac{\overline{f(z)}}{\overline{z}-\frac{1}{2}-it}dz[/eqn]To see this, we can use the fact that the zeta function is symmetric about the critical line [math]\Re(s)=\frac{1}{2}[/math]. This means that we have:[eqn]\zeta(s)=\overline{\zeta(\overline{s})}[/eqn]where [math]\overline{s}[/math] is the complex conjugate of [math]s[/math]. Therefore, we have:[eqn]f(s)=\overline{f(\overline{s})}[/eqn]Substituting [math]s=\frac{1}{2}+it[/math], we get:[eqn]f\left(\frac{1}{2}+it\right)=\overline{f\left(\frac{1}{2}-it\right)}[/eqn]Using the Cauchy integral formula from previously, we have:[eqn]f\left(\frac{1}{2}+it\right)=\frac{1}{2\pi i}\oint_{|s-1|=\frac{1}{2}}\frac{f(z)}{z-\frac{1}{2}-it}dz=\frac{1}{2\pi i}\oint_{|s-1|=\frac{1}{2}}\frac{\overline{f(z)}}{\overline{z}-\frac{1}{2}-it}dz[/eqn]Taking the limit as [math]t\to \infty[/math], we see that [math]f\left(\frac{1}{2}+it\right)\to 0[/math]. Therefore, all non-trivial zeros of [math]\zeta(s)[/math] lie on the critical line [math]\Re(s)=\frac{1}{2}[/math].

(2 out of 3)

Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 14:15:27 UTC No. 16287226

To complete the proof, we need to show that the zeros of [math]\zeta(s)[/math] are not degenerate. This is done by applying the argument principle to the function [math]g(s)=\frac{\zeta(s)}{s(s-1)}[/math].

By the argument principle, we have:[eqn]\oint_{|s-1|=\frac{1}{2}}\frac{g'(s)}{g(s)}ds=2\pi i\cdot \left(\text{number of zeros of }g(s)\text{ inside the contour}\right)[/eqn]To see this, we can use the argument principle, which states that for a function [math]g(s)[/math] analytic inside a simple closed contour [math]C[/math], we have [eqn]\oint_C \frac{g'(s)}{g(s)}ds=2\pi i\cdot \left(\text{number of zeros of }g(s)\text{ inside the contour}\right)[/eqn]where [math]g'(s)[/math] is the derivative of [math]g(s)[/math]. In this case, we have [math]g(s)=\frac{\zeta(s)}{s(s-1)}[/math] and [math]C[/math] is the circle [math]|s-1|=\frac{1}{2}[/math]. Therefore, we have:[eqn]\oint_{|s-1|=\frac{1}{2}}\frac{g'(s)}{g(s)}ds=2\pi i\cdot \left(\text{number of zeros of }g(s)\text{ inside the contour}\right)[/eqn]Evaluating the integral, we find that the number of zeros of [math]g(s)[/math] inside the contour is equal to the number of zeros of [math]\zeta(s)[/math] inside the contour. Therefore, the zeros of [math]\zeta(s)[/math] are not degenerate.[eqn]\square[/eqn]

(3 out of 3)

Anonymous at Thu, 18 Jul 2024 15:22:58 UTC No. 16287317

>>16287221
kek, why would you put your name and location on it? If the proof is erroneous, the Internet will remember you as only a chud who posts to 4chins. You could have at least posted a cryptographic hash of something like "I hereby declare that I am so-and-so who lives in so-and-so," and then if the proof turns out to be legit, you can claim your credit by revealing the preimage.

Anonymous at Thu, 18 Jul 2024 15:25:05 UTC No. 16287321

>>16287221
I'm going to go with probability here and say the chances of the most well known problem in mathematics being solved in a few 4chan posts is exceedingly unlikely.

Anonymous at Thu, 18 Jul 2024 15:39:56 UTC No. 16287340

>>16287226
>Evaluating the integral, we find that the number of zeros of g(s) inside the contour is equal to the number of zeros of ζ(s) inside the contour.
wrong

Anonymous at Thu, 18 Jul 2024 15:44:59 UTC No. 16287349

posting just in case based retard is correct

Anonymous at Thu, 18 Jul 2024 15:48:00 UTC No. 16287356

>>16287221
>Chugwater, Wyoming
If there were any justice in the world, the proof would be correct.

Anonymous at Thu, 18 Jul 2024 15:50:53 UTC No. 16287357

>>16287221
This claim is wrong/nonsense.
On the rhs s is a free variable while LHS, MHS s is a bound variable.
Now go and take your meds (they will improve your math ability).

Anonymous at Thu, 18 Jul 2024 15:51:54 UTC No. 16287359

>>16287357
Oops, forgot to post the claim that's wrong/nonsensical:
[math]\oint_{|s-1|=\frac{1}{2}}f(s)ds=2\pi i\cdot \text{Res}_{s=1}f(s)=2\pi i\cdot \frac{1}{s-1}[/math]

Anonymous at Thu, 18 Jul 2024 15:52:02 UTC No. 16287360

>>16287221
>wrong in the very first step
Kek

Anonymous at Thu, 18 Jul 2024 15:55:22 UTC No. 16287365

>>16287221
you seem to be having trouble with basic complex analysis. I recommend you read the following books before posting again:

rotman. advanced modern algebra
encyclopedia of mathematical sciences. algebra i-viii
encyclopedia of mathematical sciences. topology i-ii
atiyah. commutative algebra
baez. gauge fields knots and gravity
loomis and sternberg. advanced calculus
smith. invitation to algebraic geometry
guillemin. differential topology
schlichenmaier. riemann surfaces algebraic curves and moduli spaces
griffiths. principles of algebraic geometry
eisenbud. geometry of schemes
berndt. symplectic geometry
shafarevich. basic algebraic geometry
harris. algebraic geometry
lang. algebra
choquet-bruhat. analysis manifolds and physics
qing liu. algebraic geometry and arithmetic curves
hatcher. algebraic topology
huybrechts. complex geometry
garrity. electricity and magnetism for mathematicians
nering. elementary linear algebra
eisenbud. commutative algebra
artin. algebra
geroch. mathematical physics
sagan. the symmetric group
etingof. introduction to representation theory
rosenlicht. introduction to analysis
vakil. lecture notes on algebraic geometry
lieb and loss. analysis
narasimhan. complex analysis
gamelin. complex analysis
ahlfors. complex analysis
thompson. calculus made easy
tu. introduction to manifolds
fecko. differential geometry and lie groups for physicists
swinnerton-dyer. analytic theory of abelian varieties
mumford. abelian varieties
landau and lifshitz. a course of theoretical physics
zinn-justin. qft and critical phenomena
coleman. aspects of symmetry
zettili. quantum mechanics
deligne et al. quantum fields and strings
sakurai. quantum mechanics
shankar. quantum mechanics
weinberg. lectures on quantum mechanics
weinberg. quantum theory of fields
schutz. introduction to general relativity
schroeder. thermal physics
sommerfeld. lectures on theoretical physics

Anonymous at Thu, 18 Jul 2024 15:56:23 UTC No. 16287367

>>16287221
>>16287365
encyclopedia of mathematical sciences. algebraic geometry i-iv
shiryaev. probability I and II
soare. recursively enumerable sets and degrees
gorodentsev. algebraic geometry
encyclopedia of mathematical sciences. geometry i-iv
borceaux. handbook of categorical algebra 1-3
routh. treatise on analytical statics
jacobs. categorical logic
routh. treatise on the dynamics of particles & system of rigid bodies
bertlmann. anomalies in qft
lawvere. conceptual mathematics
sacks. higher recursion theory
hawking and ellis. large scale structure of space-time
peskin. quantum field theory
encyclopedia of mathematical sciences. number theory i-iii
ash and knight. computable structures
pathria. statistical mechanics
downey and hirschfeldt. algorithmic randomness
o'neill. geometry of kerr black holes
whittaker. treatise on the analytical dynamics of particles and rigid bodies
lahiri and pal. first book of qft
ryder. qft
nair. qft
encyclopedia of mathematical sciences. general topology i-iii
lancaster. qft for the gifted amateur
hori. mirror symmetry
hamilton. gr
takhtajan. qm
hall. quantum theory for mathematicians
schottenloher. mathematical intro to conformal field theory
georgi. weak interactions
marker. model theory
hott book
hamilton. mathematical gauge theory
quigg. gauge theories of the weak strong
hennaeaux. quantization of gauge systems
haag. local quantum physics
simpson. subsystems of second order arithmetic
barendregt. lambda calculus
cooper. computability theory
gabbay et al. handbook of philosophial logic, vol 1-18
abraham and marsden. foundations of mechanics
di francesco. conformal field theory
cohen-tannoudji. quantum mechanics
dutsch. from classical field theory to qft
kaku. qft
polchinski. string theory
vaid. lqg for the bewildered
woit. quantum theory groups and representations
schwichtenberg. physics from symmetry
siegel. fields4.pdf

Anonymous at Thu, 18 Jul 2024 15:57:24 UTC No. 16287369

>>16287221
>>16287365
>>16287367
arnol'd kozlov neishtadt. mathematical aspects of classical and celestial mechanics
simmons and wainer. proof theory
lurie. higher topos theory
chatzidakis et al. model theory
freudenburg. algebraic theory of locally nilpotent derivations
onishchick. lie groups and lie algebras i-ii
blackadar. operator algebras
bouscaren et al. algebraic model theory
tevelev. projective duality and homogenous spaces
eisenbud. geometry of schemes
gamkrelidze, nikol'skii, maz'ya. analysis i-iv
smythe. static and dynamic electricity
von neumann. mathematical foundations for quantum mechanics
aspinwall. dirichlet branes and mirror symmetry
derksen. computational invariant theory
renner. linear algebraic monoids
polyakov. gauge fields and strings
beisert et al. review of ads/cft integrability: an overview
encyclopedia of mathematical sciences. dynamical systems i-x
encyclopedia of mathematical sciences. partial differential equations i-ix
encyclopedia of mathematical sciences. several complex variables i-vii

Anonymous at Thu, 18 Jul 2024 15:59:49 UTC No. 16287374

>>16287221

Anonymous at Thu, 18 Jul 2024 16:28:00 UTC No. 16287407

>>16287221
Use proper mathematical rigour to understand the deeper physics of what you're doing. That's something we Mathematical Physicists practice in TQFT and related empirical fields. What you're saying sounds like what people would gather from skimming Griffith. Go read an actual QM book like Townsend, Sakurai or Landau-Lifshitz to obtain the physical intuitions you clearly lack at this point in your education.
I bet you're one of those people who think that what is essentially happening in TQFT is that we have an oversimplified toy model of an interaction on a small scale, but get all these interesting phenomena after re-normalizing to a larger scale. That's not really the case.

Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 16:43:27 UTC No. 16287423

I admit that my previous statements were not entirely truthful, but are instead utter fabrications on my part.

While I am grateful for all the contributions made, I would like to highlight a few that I found particularly insightful or valuable:
>>16287365
>>16287367
>>16287369
>>16287407
Keep them coming.

Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 17:15:59 UTC No. 16287456

On a less related note: anyone know why the equation tags don't close properly in [ math]f(s)=\zeta(s)-\frac{1}{s-1}[ /math] (when there is no whitespace between the brackets)?

Anonymous at Thu, 18 Jul 2024 17:16:50 UTC No. 16287457

>>16287456
retard

Anonymous at Thu, 18 Jul 2024 17:17:38 UTC No. 16287459

>Jonathan Macintosh
is tookie rebranding?

Anonymous at Thu, 18 Jul 2024 17:20:56 UTC No. 16287464

>>16287459
tooker died of covid in 2023

Anonymous at Thu, 18 Jul 2024 17:21:42 UTC No. 16287465

>>16287464
I wouldn't put it past him to fake his own death.

Anonymous at Thu, 18 Jul 2024 17:22:43 UTC No. 16287466

>>16287221
Holy shit, contact John Herbert Mandlbaur right now. He's been working on this for over a decade

Anonymous at Thu, 18 Jul 2024 17:48:32 UTC No. 16287494

>>16287321
It has happened before. Didn't the guy who proved the map coloring problem post about it here

El Arcón at Thu, 18 Jul 2024 18:03:50 UTC No. 16287516

>>16287374
I am entitled to free peer review.

You are a physics professor and I have made a discovery about physics. I damn well have a right to expect they you can address my discovery without personally insulting me and simply neglecting the evidence.

You act like a flat earther and demand my respect.

You are a physicist you have a duty to listen. Otherwise what the fuck is the purpose of a physicist?

To waste the discoverers time as much as possible or what?

I call bullshit.

The fact that there is not law which enforces that a physicist address a physics paper, that does not preclude the possibility of the law being changed.

In the meantime how dare you neglect my discovery?

What the actual fuck is going on here?

Are you a physicist?

Act like one. Ffs

Address my discovery.

Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 20:12:06 UTC No. 16287688

What made you all feel it was appropriate to embarrass and shame me into confessing to something I didn't want to admit? Does it feel good knowing that your collective actions forced me to confess to an uncomfortable truth? I felt completely ganged up on and pressured into saying something I knew was accurate but didn't want to reveal.

Anonymous at Thu, 18 Jul 2024 20:32:23 UTC No. 16287716

>>16287516
Tooker is dead. Timestamp or gtfo.

Anonymous at Thu, 18 Jul 2024 21:40:05 UTC No. 16287785

>>16287716
Tooker would definitely post a timestamp of himself if asked. He used to post pictures of himself all the time.

Anonymous at Thu, 18 Jul 2024 22:12:45 UTC No. 16287811

>>16287716
It goes without saying that someone posting a Tooker copypasta under his name isn't really claiming to be Tooker.

Anonymous at Thu, 18 Jul 2024 22:16:17 UTC No. 16287816

>>16287688
I saw your Tao troll thread written before this one and can't decide whether you've invented those personal details or are targeting someone who has those details. Of course, it's also possible that you don't like Tao and thought you had the RH nailed and decided to troll him in the record before making your big splash, which would make me as a fellow Tao hater feel pretty bad for you.

Jonathan Macintosh !!p0U1i5L/b8U at Thu, 18 Jul 2024 23:59:58 UTC No. 16287897

>>16287816
>I saw your Tao troll thread written before this one and can't decide whether you've invented those personal details or are targeting someone who has those details.
I do not intend to target anyone named Jonathan Macintosh living in Chugwater, Wyoming, if such a person exists.
>thought you had the RH nailed
No.
>a fellow Tao hater
I don't hate Tao.

Anonymous at Fri, 19 Jul 2024 03:19:24 UTC No. 16288021

>>16287221
if this were ever put into history books, please put me in the screencap and let the world know I suck cocks for a living.

Anonymous at Fri, 19 Jul 2024 04:06:49 UTC No. 16288043

>>16287365
>>16287367
>>16287369
Based