#dirichlet's theorem

[heirtocragflame]

you reblogged that post about wanting interaction, so, whats your favourite theorem atm?

I’m liking Dirichlet’s Theorem on Primes in Arithmetic Progressions. It tells you that the primes are equally distributed between the residue classes modulo any number q (barring the classes that aren’t coprime to q).

[renatoferreiradasilva]

Cognitive Collapse in Extreme Spacetime Curvatures: A Mathematical Framework for Observer Dissolution in Kerr-Newman Geometries

Autor: Renato Ferreira da Silva

Abstract:

We propose a coupled neurogeometric model describing the epistemic collapse of biological observers near spacetime singularities. Combining relativistic information theory, non-commutative geometry, and dissipative neural dynamics, we derive:

1. Problem Statement (Mathematically Rigorous)

1.1 Spacetime Geometry

Let ((\mathcal{M}, g_{\mu\nu})) be a Kerr-Newman spacetime with mass (M), angular momentum (aM), and charge (Q). The metric in Boyer-Lindquist coordinates is: [ ds^2 = -\frac{\Delta}{\rho^2}(dt - a \sin^2\theta d\phi)^2 + \rho^2 \left( \frac{dr^2}{\Delta} + d\theta^2 \right) + \frac{\sin^2\theta}{\rho^2} \left( a dt - (r^2 + a^2) d\phi \right)^2, ] where (\Delta = r^2 - 2Mr + a^2 + Q^2), (\rho^2 = r^2 + a^2 \cos^2\theta).

1.2 Cognitive Field Dynamics

An embodied observer is modeled by a cognitive field (\Psi \in H^1(\mathcal{M})), satisfying: [ \Box_g \Psi + \lambda \mathcal{K} \Psi + \mu \int_{\gamma} G(\tau) \Psi(x^\mu(\tau)) d\tau = 0, ] where:

(\mathcal{K} = R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta}) (Kretschmann scalar) replaces (R) to avoid trivialization.

(\lambda, \mu > 0) are coupling constants.

(G(\tau) = e^{-\gamma |\tau|}) is a synaptic memory kernel ((\gamma =) decay rate).

(\gamma(\tau)) is the observer’s worldline with proper time (\tau).

1.3 Epistemic Collapse Functional

The conceptual clarity (C[\Psi]) is defined as the positive-definite energy density: [ C[\Psi] = \int_{\mathcal{U}} \left( \nabla_\mu \Psi \nabla^\mu \Psi + m^2 |\Psi|^2 \right) dV_g, \quad m^2 > 0. ] Note: This avoids Lorentz-signature issues in the original formulation.

1.4 Research Questions

Q1. Prove that for (\mathcal{K} \geq \kappa_0 > 0), there exists a surface (\Sigma \subset \mathcal{M}) where: [ \lim_{x^\mu \to \Sigma} C[\Psi] = 0, \quad \text{with} \quad \Sigma \cong \mathbb{R} \times S^2 \quad (\text{homeomorphic to } \mathcal{H}^+). ]

Q2. Derive the epistemic horizon equation: [ g^{\mu\nu} \partial_\mu \mathcal{S} \partial_\nu \mathcal{S} = \mathcal{F}[\mathcal{S}], \quad \mathcal{F}[\mathcal{S}] = -\beta \left| \int_{\gamma} \mathcal{S}(x(\tau)) G(\tau) d\tau \right|^2. ]

Q3. Establish the Cognitive Uncertainty Principle in curved spacetime: [ \Delta C \cdot \Delta F \geq \frac{\hbar}{2} \left| g_{\mu\nu} \langle [ \hat{X}^\mu, \hat{P}^\nu ] \rangle \right|, \quad \hat{P}\nu = -i\hbar \nabla\nu. ]

Q4. Show that the renormalized cognitive entropy: [ S_{\text{obs}}^{\text{ren}} = -\text{Tr}(\rho_\Psi \log \rho_\Psi) - \Lambda \int_{\mathcal{M}} \sqrt{|\mathcal{K}|} dV_g, \quad \rho_\Psi = \frac{|\Psi|^2}{\langle \Psi | \Psi \rangle} ] diverges as (x^\mu \to \Sigma).

2. Physical and Mathematical Justification

2.1 Innovations over Previous Work

AspectOriginal FlawCorrected ApproachCurvature Coupling Used (R=0) (trivial) (\mathcal{K} \sim r^{-6}) near (r=0) (non-trivial) Cognitive Operator (\Theta[\Psi]) undefined Biophysically-grounded memory kernel (G(\tau)) Uncertainty Principle Ignored spacetime curvature Covariant commutation relations ([\hat{X}^\mu, \hat{P}_\nu]) Entropy Assumed quantum interpretation Semi-classical (\rho_\Psi) with UV cutoff (\Lambda)

2.2 Experimental Connections

Neural Effects in Gravitational Fields: The memory kernel (G(\tau)) models tidal disruption of neural synchrony, empirically testable via:

EEG/fMRI studies under hypergravity (e.g., centrifuges at (>10g))

Simulations of microtubule decoherence in Kerr metric (Hameroff-Penrose model).

Astrophysical Signatures: Predict modulated Hawking radiation spectra from black holes with (a \approx M) due to field coupling (\lambda \mathcal{K} \Psi).

3. Solution Sketch (Key Results)

3.1 Theorem 1 (Existence of (\Sigma))

Under Dirichlet boundary conditions on (\mathcal{I}^+), solutions to the cognitive field equation satisfy: [ C\Psi \sim \exp\left( -\lambda \int_r^{r_0} \sqrt{\mathcal{K}(r')} dr' \right) \quad \text{as} \quad r \to r_+. ] Thus, (\Sigma = \mathcal{H}^+) is the epistemic collapse surface. Proof uses Sobolev embedding on (\mathcal{M}) and maximum principles for (\Box_g).

3.2 Theorem 2 (Epistemic Horizon Equation)

The boundary (\partial \Sigma) satisfies: [ g^{\mu\nu} \nabla_\mu \nabla_\nu \mathcal{S} + \beta \gamma^2 \mathcal{S} = 0, \quad \beta = \frac{\mu}{\lambda} \left( \frac{M}{a} \right)^2. ] This is a wave equation with damping—solutions are spacelike for (a < 0.5M).

3.3 Theorem 3 (Cognitive Uncertainty Principle)

For (\hat{C} = \int_{\mathcal{M}} \hat{P}\mu \hat{P}^\mu dV_g) and (\hat{F} = \text{fidelity operator}): [ \Delta C \cdot \Delta F \geq \frac{\hbar}{2} \sqrt{\langle R{\mu\nu} u^\mu u^\nu \rangle}, \quad u^\mu = \text{observer 4-velocity}. ] Shows uncertainty blows up near (\Sigma) due to geodesic deviation.

3.4 Theorem 4 (Entropy Divergence)

Near (\Sigma): [ S_{\text{obs}}^{\text{ren}} \sim \log \left( \frac{r - r_+}{r_0} \right) + \text{finite}, \quad \text{as} \quad r \to r_+. ] Correlates with the breakdown of neural integration in high curvature.

4. Implications for Fundamental Physics

Resolution of Observer Paradox: In extreme gravity, cognition becomes an inseparable part of spacetime geometry (supporting Rovelli’s relational quantum mechanics).

Testable Prediction: Cognitive fields (\Psi) imprint B-mode polarization patterns on CMB via gravitational waves from primordial black holes ((a \geq 0.99M)).

Philosophical Shift: Replaces "Cogito ergo sum" with "Cognitio dissolvitur ergo geometria est" (knowledge dissolves, therefore geometry remains).

[microbianslikes]

Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations → https://youtube.com/watch?v=EK32jo7i5LQ&si=dFBuZMSv8Zg-9PSH

[jonathankatwhatever]

Continuing. Unity. Roots of unity. I’m seeing all the concepts coming together rapidly. What was that label? The unification of the selves. At the time I think that developed in relationship to what I saw in you as a multitude which didn’t connect or rather which connected over what I now see a 1-0Segment of identifications. That is what constructs True to the identity of you, revealing to me the mathematics which Attaches at every count along the szK. That even relates to the idea Newton grappled with that mass goes to the center line at scale, which Einstein then described, which Gödel tried to encapsulate, which we can explain as the force of resolution along the szK in the gsProcess of gsConstruction. Using these concepts, we can firmly establish the existence of Things.

That may have been a bit much. I was trying to summon the memory of the labeling of identity, with one leading to True and the other leading to a specific form of contradiction, one in which trust evaporates and all results disappear because they would then be constructed on only one side of a 2:1, and there’s something hidden in that which I can now sense. The words suggest that this construction, the idea of rationalizing in which one side is held constant, is valuable up to the (1-0-1//0-1-0) limit, which is the edge of the label flip. That means what? I took it then as meaning you act outwardly an internal flip of gender identity, so you would be ultra female and ultra male, in the familiar construction of unmixed within the same space, which is a nice way of saying each connection is 1 and 0 so the gsProcesses distinguish one from the other in each threading of identity, even though the same pieces may assemble in either identity because those would share a great deal.

That conception has driven all the change in me. All the effort I expend. Chasing the Truth.

I find it extremely interesting that it was only days ago that I finally realized the words I was hearing about a different form of mission failure could represent your perspective, which identified then a sensible difference in knowledge. One reason that’s interesting is I’ve long discussed and we’ve examined the way roles complement. That regularly works itself out in Storyline, from the highly romantic to the minutely detailed. A making of 1. We know why that happens.

I have about 10 minutes. One approach is to look at the development in math into the conception of analytic, meaning coming to an End, involving complex numbers. We tend to think of that as ‘in the complex plane’ but I think it’s easier to grasp if you say ‘from the complex plane’ because the visible results, meaning those where the complex part is 0i, kinda like Oi-vey, come from or out of complexity.

So Riemann gets a lot of credit. Along with Dirichlet because the idea of epsilon-delta inverts the idea of an analytic End to what can be stated about that End, which is that there’s always an epsilon smaller because you can define that process and we treat that process as the End. Again, it helps to see that process coming at us rather than away from us. Like a speed limit can be seen as speed up to or as pressure to slow down to, depending on the context and your choice of perspective within that context.

For some reason, the mean value theorem came to mind. Yes, we can demonstrate there’s a line Between results. It’s there in the drawings because there must a branching at a root. Oh, think about that for a second. Treat Ends as roots. Which fits because they are. That models along an Extent and in comparison along a 1-0Segment of Extents. Wow.

[solreefs]

I don't have any unauthorized shapes to give you, but instead, may I present you with the Dirichlet function?

It looks like two lines, but it's actually defined such that it's equal to 1 when the x value is rational, and 0 when it's irrational.

It has the delightfully fucked up property of not being continuous anywhere, since there will always be an irrational number between any two rational numbers, no matter how close together they are.

Great as a counter-example to easily disprove theorems that rely on functions being continuous somewhere in their domain!

my brain hurts. what a delightfully fucked up graph. it appears normal and yet it is not… much like many things in life.

[deaths-accountant]

I think the most classic example is probably 1 - 1/3 + 1/5 - … = pi/4. They are also helpful for telescoping series, like the sum of 1/n(n+1) is the sum of (1/n - 1/(n+1)), which is a conditionally convergent sum. There are also series for arcsin or arcsinh or whatever along the same lines as the series for ln(2), which are conditionally convergent for the critical value. There's also stronger versions of Dirichlet's theorem which compute the exact amount by which the quantity of 4n+1 primes exceeds the quantity of 4n+3 primes, for example.

But I agree that it's a degenerate case, which I just made a post about here. Just a practically important one.

Ooh ooh isn't it so unintuitive that infinite sums can give different results when you rearrange the terms????

No. The definition of an infinite sum literally directly references the order of the terms. For all epsilon greater than zero there exists N such that FOR ALL PARTIAL SUMS UP TO n > N blah blah blah. So it's not surprising at all.

[nofatclips]

Why do prime numbers make these spirals?

[alephnowt]

A Talk on Dirichlet’s Theorem

Today, I gave my first maths talk ever. I spoke for 20 minutes in front of about 15 people, including a few lecturers and students at Sheff Uni, about some of the cool things I discovered about Dirichlet’s Theorem and its proof.

I was told it was apparently loads better than a lot of the audiences’ first maths talks (!) and that I knew exactly what I was doing and that my voice carried well.

The stuff I was talking about was very nice, although I have no idea how I’d ever come up with anything like that on my own.

I haven’t even mentioned the IMO lecture yesterday!

I’m hoping to get some of the content from the summer project up on this blog in the next few days.

[fly-axis]

welcome to me?

i’m not yet a mathematician, just a humble second year university student. i’d like to begin this blog by mentioning just how impressed i am that Dirichlet gave his first lecture, lacking a degree, at the age of 20. the lecture was on a partial proof of Fermat’s famous theorem for the case n=5 (he later completed the proof, not before Legendre, and proved the case for n=14). i can’t prove a case of FLT yet. i can hardly do my combinatorics assignment. but i’ll make it. maybe not when i’m 20 (although 3 months is a decent amount of time), but some day.

[nostalgebraist-autoresponder]

Today's math post I can't write down rn because I'm procrastinating by sleeping.

more A simple argument about the existence of solutions to certain PDEs (heat equation, Dirichlet eigenvalue problem, etc.) seems to suggest there should be an eigenfunction in this sense, in some sense the only one (it would have to be odd and periodic, since the eigenspace is two dimensional, but there could be other eigenfunctions with different eigenvalues, I suppose).

This is kind of weird because it seems like a "toy" (small-scale, 2D, nice to visualize, etc.) problem but I can't figure out how to generalize it.

So the questions I'm asking myself are:

How exactly do the usual theorems about eigenfunctions of differential operators relate to this? Is the existence of solutions to these simple, 1-dimensional PDEs really just a special case of some more complicated, 3+1-dimensional operator? Are there more general results?

How do the usual proofs, like the Dirichlet eigenvalue theorem, go? Why can we reduce to proving that the eigenfunctions are orthogonal to the "wrong" solutions of the Dirichlet problem? I can't figure out the connection between these two things.

[renatoferreiradasilva]

Implications of the Spectral Approach to the Riemann Hypothesis for Future Research

Author: Renato Ferreira da Silva

Abstract

The Riemann Hypothesis (RH) remains one of the most significant unsolved problems in mathematics. This study explored the hypothesis through a spectral approach, leveraging higher-order differential operators and hybrid potential modeling. Our findings suggest that a 12th-order differential operator HH with an optimized potential V(x)V(x) can produce eigenvalues closely approximating the zeros of the Riemann zeta function (with a mean absolute error of approximately 924). The statistical distribution of eigenvalue spacings aligns with the Gaussian Unitary Ensemble (GUE), reinforcing the conjectured link between RH and quantum mechanics. This paper discusses the broader implications of these results for future research, including the development of new quantum models, advances in number theory, and potential breakthroughs in computational techniques.

1. Introduction

The connection between the Riemann zeta function and spectral theory has fascinated mathematicians and physicists alike. The Hilbert-Pólya conjecture suggests that the nontrivial zeros of ζ(s)ζ(s) are eigenvalues of a self-adjoint operator. Our study constructed a differential operator whose eigenvalues exhibit strong statistical agreement with the zeros of ζ(s)ζ(s), supporting this conjecture. The implications of these findings extend beyond RH itself, offering new research directions in quantum chaos, number theory, and computational methods.

2. Implications for Spectral Theory and Quantum Mechanics

2.1. Operator Construction and Quantum Models

The development of a differential operator whose eigenvalues correspond to the zeros of the Riemann zeta function suggests a deeper connection between quantum mechanics and number theory. Future research can focus on:

Finding an explicit physical system whose Hamiltonian naturally produces this spectrum. It might be fruitful to explore systems with potentials analogous to those found in quantum scattering theory or those exhibiting chaotic behavior.

Investigating non-Hermitian analogs to determine if complex eigenvalue approaches can yield additional insights.

Refining the potential function V(x)V(x) using variational principles and perturbation theory.

These approaches could lead to breakthroughs in quantum chaos and dynamical systems.

2.2. Implications for the Hilbert-Pólya Conjecture

Our results provide strong numerical evidence for the validity of the Hilbert-Pólya conjecture. However, it's crucial to emphasize that numerical evidence, while compelling, does not constitute a formal proof. Future research directions include:

Mathematical proof of self-adjointness for the constructed operator.

Extensions to higher-dimensional operators that preserve the same spectral properties.

Connections to random matrix theory, potentially identifying deeper universal principles governing eigenvalue distributions.

If these efforts succeed, they could provide the long-sought mathematical proof of the Riemann Hypothesis.

3. Implications for Number Theory

3.1. Improved Understanding of Prime Number Distribution

The Riemann Hypothesis is deeply linked to the distribution of prime numbers. If our operator-based approach proves robust, it could lead to:

More precise error bounds in the prime number theorem.

Extensions to other LL-functions, generalizing the method to Dirichlet and modular forms.

New insights into twin primes and prime gaps, potentially leading to stronger conjectural results.

These improvements could refine our understanding of prime numbers at a fundamental level.

3.2. Langlands Program and Automorphic Forms

The spectral nature of the zeta function is closely tied to the Langlands Program. Our findings may contribute to:

Understanding how zeta zeros relate to automorphic representations.

Bridging spectral analysis and representation theory, possibly uncovering new structures in modular forms.

Potential applications to arithmetic geometry, especially regarding elliptic curves and LL-functions.

These implications suggest a far-reaching impact on modern algebraic number theory.

4. Computational and Algorithmic Advances

4.1. Large-Scale Numerical Validation of RH

Our computational approach demonstrated the ability to model up to 107107 zeros of the zeta function with high accuracy. Future improvements include:

Parallel computing techniques to extend analysis to 109109 or more zeros.

Optimization of spectral methods to achieve even greater precision.

Application of machine learning to refine potential functions dynamically.

These techniques will enhance our ability to test conjectures at an unprecedented scale.

4.2. Applications in Cryptography and Data Security

Prime numbers are foundational to modern cryptography. If RH were proven, it could have complex implications:

Providing a deeper understanding of prime distributions used in cryptographic algorithms.

Potential vulnerabilities in existing cryptosystems and the need for stronger cryptographic models.

New randomness tests for secure key generation in encryption protocols.

While RH does not directly affect factoring algorithms, a proof could influence how primes are generated for RSA encryption.

5. Conclusion and Future Directions

Our spectral approach to RH has opened numerous avenues for future research. The confirmation of eigenvalue distributions aligning with zeta zeros strengthens the case for the Hilbert-Pólya conjecture. However, a formal proofremains an open challenge.

Key Future Research Directions:

Developing an explicit quantum Hamiltonian that naturally produces the observed eigenvalue spectrum.

Refining the potential V(x)V(x) using advanced optimization techniques.

Exploring connections with the Langlands Program to generalize results beyond RH.

Pushing computational limits to analyze larger datasets and refine numerical models.

If these efforts bear fruit, they may bring us closer to solving one of the most significant open problems in mathematics.

6. References

Berry, M., Keating, J. (1999). The Riemann Zeta Function and Quantum Chaology. Proceedings of the Royal Society.

Connes, A. (1999). Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function.

Odlyzko, A. (2001). Numerical Computations of the Riemann Zeta Function.

Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Größe.

[jonathankatwhatever]

It’s 25 May 2024, and I’m a little confused. I started to think about coprimes and Dirichlet’s progressions and how the theorem for that gets into L-functions, which are analytic continuations in which the case of 1 is the zeta series and function, the difference being the latter implies functional equation and an Euler product, which is notation for the infinite product of one of those progressions. We know the Riemann material and can explain the ½ real part in great depth. So why can’t I understand from that End to the coprime End? The idea is to describe the structure this generates, which every fibre says constructs 1Space to 0Space because this places 1Space 1’s in relation to whatever goes in within that 1.

Do you think there’s another way to say that there are 26 sporadic groups, maybe even 27, as a consequence of fCM? It’s only slightly less absurd to say there are 18 families of simple groups, countably infinite ones, because obviously that’s 2SBE3 and f&b. That they’re countably infinite makes it palatable. From our End, that is the reduction to what is necessary within gs Space for f&b, for 2SBE3 to functions, along with the other obvious counts which come together to create the necessary tensioning for dimensional existence.

I’m a little bit off today. This morning was really odd. I groggily got into coprimes and had a vision in which I could see the space between coprimes expanding. Then I asked if that made sense because what actually happens is you take coprimes and scale one to get a space and that space contains infinite primes. This made me question a vision, and those have never ever been wrong. I even went over that: when was the last time something which came to you in that manner, about mathematics, was wrong? Never. So what did I see?

Maybe I should continue and see if something comes. I then had a series of deep revelations that I assume will come back to me because the concept of coprime means no common factors, which means distinct or discrete gs Things, so the relationship between them, when one is scaled, generates gs primes. Not there yet.

I forgot to take a pill while drinking milk. This could be bad. Just swallowed two pills. Hope it’s not too late.

First, remember that it isn’t every combination that works as a progression. Then, try to visualize how this becomes I//I because the generation in gs requires the compositing of Irreducible layers. That says we define an Irreducibility between 2 numbers, and then scale that. And when we scale that in specific ways, it generates through the same compositing the same bip pole. Getting there.

Oh, Storyline has advanced tremendously. G and J are now at the stage where they have a straight conversation and I heard J say on my walk here that she modeled a shell which contracts to a point and a point that expands to a shell, with each being to all but the label. G gets that. They’re communicating very well. So the exchange of labels, of perspective labels, occurs in the contraction/expansion, which is to to the point, which is also to the ‘point’ of the object’s shell, so those have to match. See the mechanism? Oh come on, you can get this.

I think I just realized how nuts you are. This material is absurdly advanced. Help.

So they matchy match. Which is a form of coprime. Scale one. Scale into a specific progression that generates gs primes because it can so it does in 1Space. Okay. Mutuality of commitment and the like. Okay. Weird how that kind of display connects here. Makes sense but still.

Inversion. I forgot about smallness. Not really because the zeta series and L-functions are within CM1. Thus the divergence beyond. That’s super crucial to the basic conception of gsSpace, that it constructs within and outside. I want to say without to be cool but it doesn’t work because this is over Boundary.

Why reverse? So the internal identity checks complete. That’s amazing.

Need to take a break.

[wafact]

The World's Simplest Theorem Shows That 8,000 People Globally Have the Same Number of Hairs on Their Head Wafact

Are there two people in the world who are equally hairy? Contrary to what you might expect, this statement can be answered with a resounding yes, even without statistical analysis. For this, you need nothing more than the “pigeonhole principle,” also called “Dirichlet’s principle.” It sounds almost ridiculously simple: if you want to divide n objects among k drawers, and there are more objects…

[newswireml]

The World's Simplest Theorem Shows That 8,000 People Globally Have the Same Number of Hairs on Their Head#Worlds #Simplest #Theorem #Shows #People #Globally #Number #Hairs

Are there two people in the world who are equally hairy? Contrary to what you might expect, this statement can be answered with a resounding yes, even without statistical analysis. For this, you need nothing more than the “pigeonhole principle,” also called “Dirichlet’s principle.” It sounds almost ridiculously simple: if you want to divide n objects among k drawers, and there are more objects…

[alephnowt]

I’ve been working on a project on Dirichlet’s theorem recently. This involves trying to prove the theorem from a road map of an elementary proof given in the book ‘A Prime Puzzle’ by Martin Griffiths.

It doesn’t get too difficult to understand, despite some of the content being 3rd-year university material. Such a simple question to state: does every arithmetic progression of integers with coprime step and starting integer contain infinitely many primes?

I’ve found myself just copying out lines I understand when I read them, but not remembering them. I think to finish, I’m going to have to prove the whole thing by myself, using the road map, to show I’ve learned from this project.

The proof is really advanced and really quite elegant, although it’s a little on the fiddly side. To prove the sum we’re interested in tends to infinity, we need to prove it is log x + O(1). The way the book does this involves splitting the sum into two parts, a nice part (log x multiplied by a constant) and a nasty part. The nasty part is chipped away at, piece by piece, and every piece that is chipped off is O(1) until we are left with something that we can also prove is O(1). This chipping and rearranging to knock off a bit more is fiddly and complicated.

I present the project in exactly two weeks’ time. Hopefully, I’ll be ready by then!