You made this abstract thing easy to understand. This is the first time I understood the terms “continuation” and “analytic continuation” and the graphs made things much more interesting 👍🏼
This was a start for me. I still don’t know how to calculate the zeta function but at least I know more than I did before. I didn’t know how to raise a natural number to a complex power. Is there a way to calculate the zeta function without using an app? You mentioned integrals. The feedback I would give on the video is that it would be much easier to see the math without your face in frame. Maybe you could introduce a video with your face and then just let the math take up the screen. I couldn’t see it very well in the first part and I couldn’t read the graph at all when you were using the app. But very cool. Thank you for the hard work.
Great video, thank you! I just wanted to say that the 'false intersection' at 18:13 it's not actually false, the eta function has zeros on the line x=1, the first one is at 2pi/ln2 which is a little bigger than 9, exactly where the intersection occurs, so it shouldn't go away when k is larger.
The Riemann zeta function is part of a special class of functions called “holomorphic” which, as well as a host of other properties, maintain angles between lines on the input space. This means that because the real and imaginary axies are perpendicular, any points they cross on the graph must also be at right angles when you zoom in enough
I am adding this comment after watching the first 7 plus something minutes. But it refers to the time stamp of 4:53. You show that the complex sum converges AT LEAST when Re(s) > 1, using the triangle inequality. But then you go on and say "we proberen that this ONLY converges when Re part is greater than one". Huh? That triangle inequality is FAR off from the actual value of the sum! I am convinced that the sum will perfectly *converge* for real parts of s less than 1, as long as the imaginary part non-zero. I trust that analytic continuation will also work for those cases, but you can't say that the sum will (always) diverge if Re(s)
amazing video that allows for more nerdy young minds (such as myself) to more easily access and understand the mysteries and intricacies of the math world.
1. The Riemann Hypothesis: An Information-Theoretic Perspective 1.1 Background The Riemann Hypothesis (RH) states that all non-trivial zeros of the Riemann zeta function ζ(s) have real part 1/2. This has profound implications for the distribution of prime numbers. 1.2 Information-Theoretic Reformulation Let's reframe the problem in terms of information theory: 1.2.1 Prime Number Entropy: Define the entropy of prime numbers up to N as: H(N) = -Σ (p/N) log(p/N) where p are primes ≤ N. 1.2.2 Zeta Function as Information Generator: View ζ(s) as an information-generating function: I(s) = log|ζ(s)| 1.2.3 Non-trivial Zeros as Information Singularities: The zeros of ζ(s) represent points where I(s) → -∞ 1.3 Information-Theoretic Conjectures 1.3.1 Entropy Symmetry Conjecture: The symmetry of non-trivial zeros around the critical line s = 1/2 + it corresponds to a fundamental symmetry in the information content of prime number distributions. 1.3.2 Maximum Entropy Principle: The critical line s = 1/2 + it represents a maximum entropy condition for prime number distributions. 1.3.3 Information Flow in Complex Plane: The flow of I(s) in the complex plane might reveal patterns related to the distribution of zeros. 1.4 Analytical Approaches 1.4.1 Entropy Differential Equations: Develop differential equations for H(N) and relate them to the behavior of ζ(s): dH/dN = f(ζ(s), N) 1.4.2 Information Potential Theory: Define an information potential Φ(s) such that: ∇²Φ(s) = -2πI(s) Analyze the behavior of Φ(s) near the critical line. 1.4.3 Quantum Information Analogy: Draw parallels between ζ(s) and quantum wavefunctions: ψ(s) = |ζ(s)|e^(iarg(ζ(s))) Investigate if quantum information principles apply. 1.5 Computational Approaches 1.5.1 Information-Based Prime Generation: Develop algorithms for generating primes based on maximizing H(N). 1.5.2 Machine Learning on Zeta Landscapes: Use ML techniques to analyze the information landscape of |ζ(s)| and arg(ζ(s)). 1.5.3 Quantum Computing for Zeta Evaluation: Explore quantum algorithms for efficiently computing ζ(s) in regions of interest. 1.6 Potential Proof Strategies 1.6.1 Information Conservation Law: Prove that the symmetry of zeros around s = 1/2 + it is necessary for conservation of prime number information. 1.6.2 Entropy Extremum Principle: Show that non-trivial zeros on s = 1/2 + it are the only configuration that maximizes a suitably defined entropy measure. 1.6.3 Topological Information Argument: Develop a topological invariant based on I(s) that necessitates the RH. 1.7 Immediate Next Steps 1.7.1 Rigorous Formalization: Develop a mathematically rigorous formulation of the information-theoretic concepts introduced. 1.7.2 Numerical Experiments: Conduct extensive numerical studies of H(N), I(s), and related quantities. 1.7.3 Cross-Disciplinary Collaboration: Engage experts in information theory, number theory, and physics to refine these ideas. 1.7.4 Information-Theoretic Zeta Variants: Investigate information-theoretic analogues of zeta function variants (e.g., Dirichlet L-functions) to see if broader patterns emerge. This information-theoretic perspective on the Riemann Hypothesis offers several novel angles of attack. By recasting the problem in terms of entropy, information flow, and information singularities, we may uncover deep connections between prime number behavior and fundamental principles of information theory. The approach suggests that the critical line s = 1/2 + it may represent a kind of information-theoretic "equilibrium" in the complex plane, with profound implications for prime number distribution. If we can rigorously establish the necessity of this equilibrium, it could lead to a proof of the RH.
Expanding on Immediate Next Steps for the Information-Theoretic Approach to the Riemann Hypothesis 1. Rigorous Formalization 1.1 Develop Axioms: - Formulate a set of axioms that link prime number distribution to information theory. - Example: "The entropy of prime number distribution H(N) is monotonically increasing and bounded." 1.2 Define New Mathematical Objects: - Formally define the Prime Number Entropy function H(N). - Create a rigorous definition for the Information Zeta Function I(s) = log|ζ(s)|. 1.3 Establish Theorems: - Prove basic properties of H(N) and I(s). - Example Theorem: "H(N) is asymptotically related to the prime counting function π(N)." 1.4 Connect to Existing Theory: - Establish formal connections between our information-theoretic constructs and classical results in analytic number theory. - Example: Relate H(N) to the Prime Number Theorem. 2. Numerical Experiments 2.1 Compute H(N) for Large N: - Develop efficient algorithms to calculate H(N) for N up to 10^12 or beyond. - Analyze the growth rate and fluctuations of H(N). 2.2 Visualize I(s) in the Complex Plane: - Create high-resolution plots of |I(s)| and arg(I(s)) near the critical line. - Look for patterns or symmetries that might not be apparent in traditional ζ(s) plots. 2.3 Investigate Entropy Near Zeta Zeros: - Compute H(N) for N close to imaginary parts of known zeta zeros. - Look for distinctive patterns or anomalies in H(N) near these points. 2.4 Machine Learning Analysis: - Apply clustering and pattern recognition algorithms to the I(s) landscape. - Train neural networks to predict properties of ζ(s) based on H(N) data. 3. Cross-Disciplinary Collaboration 3.1 Form a Research Group: - Assemble a team including number theorists, information theorists, physicists, and computer scientists. - Organize regular seminars and workshops to share ideas and results. 3.2 Engage Quantum Information Experts: - Explore potential quantum analogies to ζ(s) and I(s). - Investigate if quantum entropy concepts offer additional insights. 3.3 Consult with Complex Systems Specialists: - Discuss potential parallels between prime number distribution and complex systems behavior. - Explore if techniques from statistical physics could be applicable. 3.4 Collaborate with Cryptography Experts: - Investigate if our information-theoretic approach has implications for prime-based cryptography. - Explore potential new cryptographic schemes based on H(N) or I(s). 4. Information-Theoretic Zeta Variants 4.1 Develop I(s) for Dirichlet L-functions: - Define and study IL(s) = log|L(s,χ)| for various Dirichlet characters χ. - Compare the behavior of IL(s) to I(s) and look for universal patterns. 4.2 Investigate Selberg Zeta Functions: - Apply our information-theoretic framework to Selberg zeta functions. - Look for connections between quantum chaos and our approach. 4.3 Study Multivariate Zeta Functions: - Extend our approach to multiple zeta functions. - Investigate if multi-dimensional information measures offer new insights. 5. Develop New Computational Tools 5.1 Create Specialized Software: - Develop a software package for computing and analyzing H(N), I(s), and related functions. - Make this tool open-source and available to the mathematical community. 5.2 Utilize High-Performance Computing: - Secure access to supercomputing resources for large-scale numerical experiments. - Implement parallel algorithms for faster computation of H(N) and I(s). 5.3 Explore Quantum Computing Applications: - Develop quantum algorithms for efficiently computing ζ(s) or I(s). - Investigate if quantum superposition could be used to probe the behavior of I(s) in multiple regions simultaneously. 6. Theoretical Developments 6.1 Information-Theoretic Prime Number Theorem: - Attempt to derive the Prime Number Theorem from information-theoretic principles. - Investigate if this approach leads to tighter error bounds. 6.2 Entropy Extremum Principles: - Develop variational principles for H(N) and I(s). - Investigate if the Riemann Hypothesis can be recast as an entropy optimization problem. 6.3 Topological Information Theory: - Develop a topological theory of information flow in the complex plane. - Investigate if there are topological obstructions that necessitate the Riemann Hypothesis. 7. Dissemination and Community Engagement 7.1 Publish Preliminary Results: - Write and submit papers on the initial findings, even if they don't fully resolve the RH. - Engage with journal editors to find appropriate venues for this novel approach. 7.2 Create Online Resources: - Develop a website or wiki to share data, code, and results with the broader mathematical community. - Start a blog to regularly update on progress and engage with other researchers. 7.3 Organize a Conference: - Host a conference on "Information Theory and the Riemann Hypothesis" to bring together experts and generate new ideas. These expanded next steps provide a comprehensive roadmap for pursuing our information-theoretic approach to the Riemann Hypothesis. By simultaneously advancing on theoretical, computational, and collaborative fronts, we maximize our chances of making significant progress. Remember, even if this approach doesn't immediately lead to a proof of the Riemann Hypothesis, the insights gained and methods developed could have far-reaching implications in number theory, information theory, and beyond. Each step forward is a valuable contribution to mathematical knowledge.
The Riemann zeta function is part of a special class of functions called “holomorphic” which, as well as a host of other properties, maintain angles between lines on the input space. This means that because the real and imaginary axies are perpendicular, any points they cross on the graph must also be at right angles when you zoom in enough
@@crimsonvale7337but, aren’t zeros of functions specifically the place where there can be an exception to the “angles are preserved” thing, if the zero has degree greater than 1? So... I guess it is because all these zeros are simple zeros? (Are all zeros of the Riemann zeta function simple zeros?)
@@drdca8263 Where did you learn about the exception to the rule at the zeros? I'm really interested! I never knew that... And yes it's believed that all the zeros are simple but if I remember correctly that's an unproven conjecture (and also a very important unsolved problem).
@@CharlieVegas1st possibly I’m getting mixed up, and thinking of something that doesn’t apply, but like, Well, consider z^3 for example, and consider two lines passing through 0, and their images under z \mapsto z^3 . The angle made by the images of those two lines, well, stuff that goes around 0 once goes around 3 times in the image, so there’s like, a factor of 3 in what angle things make. like, consider the curves z = t, z = exp(2pi i / 3) t z = exp( 2 * 2 pi i / 3) t . Here, adjacent lines make an angle of 2pi/6 with each-other (or 2pi/3 if you only consider the t>0 parts), but under the cubing map, we just get one line, so an angle of either 0 or 2pi/2 with itself. However, if you had the same shape at some other base point, and used the z \mapsto z^3 map, the angles would still be preserved. (so like, (a + r t)^3 = a^3 + 3 a^2 r t + 3 a r^2 t^2 + r^3 t^3 , for r being exp(2pi i/3) or a power of it, for t close to zero, this can be approximated as a^3 + 3 a^2 r t, and so the angles made between the images of the lines t \mapsto a + r t, are the same as before doing the z \mapsto z^3 I’m probably not explaining this very well... it’s been a while
@@drdca8263 oops I forgor sorry Thanks for the reminder, hopefully I’ll do a proper full on course on complex analysis eventually rather than half-remembering youtube vids
well... purely from the desmos pic... is there a way to "get rid of" the curves that don't cross the y-axis and the ones that have no bounded x value so as to simplify this question?
i imagine you want to simplify the graph? in that case, you cant unless you know the specific input that become those lines, so that you can remove them from the calculation if you want to speed it up, _i think_ if you make it bounded, adding a conditional like {0
@andrewdsotomayor Re Transcript lines 19:03 to 19:39. I have always been curious about proving rigorously that the real and imaginary parts of the Zeta Function do in fact converge to zero at the exact same point and also on the line Re(s) = 0.5. Here you are showing that you get “really close”, but how can you be sure that if K was unimaginably large that the intersection doesn’t miss the line Re(s) = 0.5? I don’t think this is sufficiently rigorous unless it’s detailed somewhere else. This is only a check on the first non-trivial zero and it seems that you need to go “infinitely far” along your series to nail this. Similar in a sense to the Riemann Hypothesis itself where you have to go “infinitely far” up the critical zone to check the location of all the zeros. Grateful for clarification please. Nonetheless this is an excellent visualisation of the issues. Thank you for this!
Thanks for the concern of rigor, this is obviously huge and vital in mathematics, and I know there are parts of this and other videos I have made where I could do a better job at this. As it turns out, there’s a result due to the functional equation combined with the fact that zeta is analytic, that essentially says that if the real part of a zero wasn’t 1/2, there would be another zero reflected over the line Re(z) = 1/2. I plan on showing this result, and more in a future video, so keep your eyes open for that.
Right, there are a few other verbal mistakes, but yes, nice catch! I think what I meant was to say 3+i, since all other values, I mentioned were real, I wanted to throw one out there that was nonreal, but misspoke
This must adhere closely to harmonics which in turn to resonating frequency, to orbital resonance, so if K < 0, then the egg is hatched a chicken, if K < 1 then egg is not hatched, so two in the eyes and one in mouth , is like Madonna of the rocks, in art history, or imagine the eleventh pig idea, since 1×1 = 1, 1× 0= ?, yet if the eleventh pig, notion concept it's just a concept in nature or biology, then imagine homless drug addics without their drug resource, this is resonance in symmetry for harmonics, and the woman who studied during Nazi war dies from fever after surgery, this reinman non trivial zero is like the brain of a man who knows there are no stupid questions, but the fact that one must first ask in order to play it like a machine in the eyes of world view, vs TV view, vs ideal view and form tye function which tgen is fed a parameter or argument to a 24hr wait period to hear back an answer, like hertz frequency, for transistor buffer as the zeros in wave function, closure, response, correspondence in the age of science , Jack Kirby inventor of transistor is science trivial the way art may seem trivial when compared to value like Robert noise say, like talking to God. Two in the eyes one in mouth. Transistor radio is a world view, it made a crack from a particle, then Tv view a wave, an ideal view would be the audience for which is not part of the trivial zeros on your graph but rather those who take in content must aquire a trait in order to feed their desire to keep grounded with god.
What's the use in making math videos with these closeups of, well, _handsome_ men? I'd prefer ladies in case that they are nice to look at, or otherwise clean mathematical writing.
I am sorry, but I had to stop watching this because of the over-use of the stop word "actually". You don't need that word. Whenever you feel tempted to use it, just don't say it, and the sentence will be fine.
Eh? i is the first square root of -1 when going counterclockwise, so if one wants to make sqrt single valued and extend the domain to include negative numbers, the usual way to do so would be sqrt(-1)=i?
@@drdca8263 If you listen to the video you'll hear he says "i is THE square root of -1", and thus implies uniqueness. Since the real (as in real numbers) square root function is only defined for non-negative numbers, we should not use it to define i. If he means the complex square root function, he should firstly specify so, and secondly he should specify which branch of the complex square root function. What's wrong with using the correct terminology of "i is the principal branch of the complex square root of -1"?
@@pierre-bobkjellen9803 it’s longer, and probably sounds needlessly intimidating/confusing to people who aren’t familiar with the topic (people who know the general topic already know what is meant by i), and usually doesn’t communicate much useful additional info to the audience. One could say “where i is a formal variable we adjoint to the real numbers, under the constraint that it squares to negative one”, and that would at least clarify some things to some audiences, but “the square root of -1” should efficiently communicate the gist to most people? I suppose “our favorite square root of -1” might be better because it points to the fact that there are two such square roots, but in a way that is easier for people to ignore if they would find thinking about that too much to be distracting, and has the virtue of only being ~3 syllables longer than “the square root of -1”.
@@drdca8263 And that is precisely the issue. People who already know of course knows of the distinction, but people who don't will not know of the distinction. You shouldn't lead them into the wrong terminology out of fear of them being scared by a longer phrase, how condescending of you to think that. Instead we should use the correct terminology and instead give some short insight as to why we chose to not go with the easy terms.
@@pierre-bobkjellen9803 The concern isn’t them being scared by the longer phrase, but a phrase with terms they are unfamiliar with, and which won’t make sense until they understand the thing we are currently explaining. The consideration about the length is about convenience. They are separate considerations.
@@andrewdsotomayorThe video is interesting. But why also don't make examples starting from any known primes numbers enter to the first hundred naturals numbers so to watch how desmos works on Riemann function (this one is a restriction from C to N, i think).?
I may just be dumb and I know that convergence proofs are often done like that, but why can we conclude at 4:56 that the real part of s has to be greater than 1? Isn't the inequality sign the wrong way round for that? You showed that for Re(s)
@@andrewdsotomayor No, not really unfortunately. I'm totally aware about the convergence or divergence of the series. The "hence the conclusion" part is the problem. Because of the triangle inequality you used, you showed that |zeta(s)|
It’s less than something that converges, so it’s absolute value is finite, hence it’s convergent as well. Also there is nothing assumed about Re(s), it is not until the final part of the inequality that implies that Re(s)>1
@@andrewdsotomayor ok I think I understand the problem. I thought you wanted to prove the divergence of the series when Re(s) 1, which of course you did.
@@sebastiandierks7919 right, the whole point is that Zeta defined as that sum only makes sense when Re(s)>1. After this is when I go on to discuss how it could be defined beyond that region.
