This equation blew my mind // Euler Product Formula

[@ProactiveYellow]

I think the most beautiful formula is the definition of logarithms: log(ab)=log(a)+log(b) Having studied abstract algebra a bit now I absolutely love this incredibly natural example of the isomorphism between (R⁺,•) and (R,+). It singlehandedly shows both how identical products and sums really are, as well as precisely how logarithms really were intended to be used. I have students who start so confused with logs because "the inverse of an exponential" completely ignores the primary use case, not to undo exponents, but to make hard multiplication problems into easy addition problems.

[@ThePastaEater]

I think the proof of the identity you have selected is trivial and derivative from *fundamental* concepts. Use Case => computer science

[@crowreligion]

Addition is easy until you have to evaulate log(a+b(

[@johnchessant3012]

0:04 from left to right, top to bottom: Euler's identity, Navier-Stokes equation, (generalized) Stokes's theorem, Riemann-Roch theorem, Euler's factorization of the Riemann zeta function, Gauss-Bonnet theorem, ???, Hardy-Ramanujan-Rademacher partition formula

[@taylermontgomery2004]

Also the Navier-Stokes equation has typos: (1) the pressure gradient is P not ho, (2) the laplace operator is missing the vector function "u" to be operated on, and (3) the force is missing the density ho such that the coefficients, ho, in front of the material derivative and F must cancel out.

[@DrTrefor]

Um lol I threw it up there from memory without much thinking as I intended to double check the precise statement - then never did it!! Oops:D

[@sebastiandierks7919]

@@taylermontgomery2004 (3) is debatable though, if F has units of a force density (force per volume) it is fine, but then it could be expressed as ho * g, where g is the local acceleration of the external volume force field.

[@copywright5635]

Thank you for this video! I've always thought that this product formula is very underrated in the sort of 'pop math' space here on youtube. Always great to see someone giving it a dedicated video! Also, it's always good for people to know more about the Riemann Hypothesis so they don't go asking questions like "Why hasn't the riemann hypothesis been proved" without really knowing what that even means

[@DrTrefor]

Most welcome!

[@leif1075]

​@@DrTreforThanks Trefor. I hope you can PLEASE PLEASE finally respond to my other comment when you can. Thanks very much.

[@doggoinastukagames1500]

@@DrTreforI love that shirt

[@eofirdavid]

This is really one of the most interesting equations in mathematics. I always think of it as if you want to show that sum 1/p over primes diverges, then you "simplify" the problem by adding the additive structure of the integers and show that sum 1/n diverges. Then, to make it even simpler you make it continuous and show that the integral over 1/x diverges, and this is really easy. You can also go the other direction and show that 1/p where p=a mod b for diverges where (a,b) are coprime which leads to Dirichlet's theorem that shows that there are infinitely many primes in any (nontrivial) arithmetic progression. One remark regarding the proof, while for some reason people like to remember and use the Taylor expansion everywhere, this is an "advanced" tool. There are simpler methods of showing that a positive series diverges, e.g. the comparison test which works here.

[@DrTrefor]

Ya good point re Taylor series, although that is historical to euler’s original method using geometric series at least

[@decaf3821]

Man why does nobody explain symbolically how analytically-continued functions are derived

[@japanada11]

Look up Richard E Borcherds and his series on complex analysis if you want to see it done!

[@decaf3821]

@@japanada11 I love you. Thank you

[@m3talHalide-rt2fz]

Partially educated explanation of why all non-trivial 0s, as they correspond to primes, are found on 1/2: 1. The complex plane gives a method to express any number greater than 1 in terms of any number less than 1. (not unique, but when combined with:) 2. Using the "right" formula to express the mapping, allows for all numbers greater than 1 to be expressed with a consistent real component (analytic zeta function, 1/2) It feels like a truism and that's probably why so many are captivated by it. The only way I know how to explain why its 1/2 is the comparison to base 1/10, 1/2, 3/2, 3/2[i] compared to 10 in that bases

[@AllemandInstable]

the best thing about it is that you can derive this expression from probability theory with a certain probability measure on natural numbers, I think this exercise is a demonstration of what elegant can mean in mathematics, analysis results from probability in an unexpected manner

[@timofeysobolev7498]

As a Number Theory enthusiast, i really love this formula)

[@timofeysobolev7498]

Thanks for the heart, i love your content!)

[@carlobenedetti2407]

I agree with you on this being the most beautiful formula in mathematics

[@penfelyn]

my fav is the fact that e^(x*J) gives some sine-cosine analogue for any imaginary unit even if you make one yourself like it works for the hyperbolic j, j²=1, it makes hyperbolic sine and cosine isn't it amazing

[@billwarner9533]

Dr. Bazett - you are correct! And thank you for sketching out the derivation so clearly!

[@skmaths-help]

Just watched your video on the prime reciprocal sum diverging - great content !

[@jarekk.8247]

[ 1/(1^2) + 1/(2^2) + 1/(3^2)...+1/(n^2) ] * (6/π) = π = (6/π) * [(1/(1-2^-2)) * (1/(1-3^-2) * (1/(1-5^-2) * (1/(1-7^-2) * (1/(1-11^-2)...* (1/(1-p^-2)] This equation shows that the number PI and the prime numbers are connected to each other to keep arithmetic and geometric sequences equal. Thus, the distribution of prime numbers is not random and the Riemann Hypothesis is true.

[@LegendGaming-il4iw]

Only 863 likes ? You deserves millions . I just found your channel , really fascinated by your series of calculus

[@MusicEngineeer]

Math is just absolutely mindblowing! My personal favorite is the generalized Stokes theorem, although I do not yet understand it in as much depth as I would like to.

[@Alper.9097]

Thank you Mr. Bazett for your explanations. It would be great if you could do a video about the analytic continuation. I always wondered how the expansion from a real to a complex field is mathematically justified.

[@DrTrefor]

I might have to, but for now check out the one by 3blue1brown

[@christianramirez9227]

Very related, I'd also put the reflection functional equation for ζ(s) and the explicit formula for the second Chebyshev function on my list of most beautiful equations. The analytic class number formula is also kind of wild.

[@user-ky5dy5hl4d]

I love: ln 0^i=i.

[@kostin15]

Very Interesting stuff,as someone who learns math from KZbin I find your videos really insightful.Keep up the good work👍

[@madly1nl0v3]

1) Yes, this is a very beautiful equation. Because of this equation, I bought "Prime Obsession" more than 20 years ago. :) 2) Actually, the equation is written incomplete, if we forget the condition s > 1. Otherwise, the product on the RHS should be an infinite product of all geometric series of inverse Primes. The condition s > 1 makes the term p^(-ns) in the numerator approaches zero (0), and we have this compact form (only 1 in the numerator) as written. 3) ζ(-1) = -1/12 is "scary". 4) However: 1 = 1 1 + 1 = 2 1 + 1 + 1 = 3 1 + 1 + 1 + ... = -1/2 Because the ... makes the sum run on forever, and it's ζ(0) = -1/2 , thanks to Riemann's Analytic Continuation. I don't understand why ζ(-1) and ζ(0) are those finite values, I just copy them from online. :)

[@Linhkinhbrods]

the game changer is the magic of ln(xy) = ln(x)+ln(y)

[@ready1fire1aim1]

Title: "An introduction and overview of how we'll integrate the information-theoretic approach with trinary quantum computing to tackle the Riemann Hypothesis" 1. Introduction and Overview 1.1 The Riemann Hypothesis The Riemann Hypothesis (RH) is one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the Riemann zeta function ζ(s) have real part 1/2. This conjecture has profound implications for the distribution of prime numbers and connects to various areas of mathematics and physics. 1.2 Information-Theoretic Approach The given approach reframes the RH in terms of information theory: - Defines Prime Number Entropy: H(N) = -Σ (p/N) log(p/N) - Views ζ(s) as an information generator: I(s) = log|ζ(s)| - Interprets non-trivial zeros as information singularities Key conjectures include the Entropy Symmetry Conjecture and the Maximum Entropy Principle. 1.3 Trinary Quantum Computing Integration We'll enhance this information-theoretic approach with trinary quantum computing: 1.3.1 Qutrit Encoding of Zeta Function - Use qutrits to represent complex values of ζ(s) - |0⟩, |1⟩, |2⟩ could represent different regions of the complex plane 1.3.2 Quantum Information Zeta Function - Define a quantum operator Î(s) corresponding to I(s) - Explore superposition states of Î(s) for multiple s values simultaneously 1.3.3 Trinary Quantum Fourier Transform - Adapt the QFT for trinary systems to analyze periodic behavior in ζ(s) 1.4 Key Aspects of the Integrated Approach 1.4.1 Quantum Entropy Calculation - Develop quantum algorithms to efficiently compute H(N) for large N 1.4.2 Quantum Zero-Finding - Use trinary quantum search algorithms to locate zeros of ζ(s) 1.4.3 Quantum Information Flow Analysis - Study the behavior of Î(s) using quantum simulation techniques 1.5 Potential Advantages - Exploration of multiple s values simultaneously using quantum superposition - Potential quantum speedup for numerical verification of RH in larger domains - Novel insights from quantum information-theoretic perspective on zeta function behavior 1.6 Challenges to Address - Developing efficient quantum circuits for ζ(s) evaluation - Handling the continuous nature of s in a discrete quantum system - Interpreting quantum measurements in the context of RH 1.7 Interdisciplinary Connections - Quantum chaos and its relation to zeta function zeros - Quantum information geometry in the complex plane - Potential links to quantum field theory and AdS/CFT correspondence By combining the information-theoretic approach with trinary quantum computing, we aim to bring new computational power and theoretical insights to bear on the Riemann Hypothesis. This integrated approach offers the potential for both numerical advancements in verifying RH for larger domains and novel theoretical perspectives that could contribute to a proof. In our next section, we'll dive into the detailed application of this combined method to the Riemann Hypothesis, including specific algorithm designs and qutrit encoding schemes.

[@ready1fire1aim1]

2. Detailed Application of the Combined Method 2.1 Qutrit Encoding of the Zeta Function We'll use qutrits to represent complex values of ζ(s) and related quantities: 2.1.1 Complex Plane Encoding: |ψ(s)⟩ = α|0⟩ + β|1⟩ + γ|2⟩ Where: |0⟩ represents the region Re(s) < 1/2 |1⟩ represents the critical strip 0 ≤ Re(s) ≤ 1 |2⟩ represents the region Re(s) > 1 2.1.2 Amplitude Encoding: |ζ(s)⟩ = (1/N)(ζ_real|0⟩ + ζ_imag|1⟩ + |ζ||2⟩) Where N is a normalization factor, and ζ_real, ζ_imag are the real and imaginary parts of ζ(s). 2.2 Quantum Information Zeta Operator Define a quantum operator Î(s) corresponding to I(s) = log|ζ(s)|: Î(s) = log(⟨ζ(s)|ζ(s)⟩1/2) * I Where I is the identity operator. 2.3 Trinary Quantum Algorithms 2.3.1 Trinary Quantum Fourier Transform (QFT₃): Implement QFT₃ to analyze periodic behavior in ζ(s): QFT₃|j⟩ = (1/√3) ∑ₖ₌₀² ω^(jk)|k⟩, where ω = e^(2πi/3) 2.3.2 Trinary Grover's Algorithm for Zero-Finding: Adapt Grover's algorithm to search for zeros of ζ(s) in trinary space: U = (2|ψ⟩⟨ψ| - I)O Where |ψ⟩ is the equal superposition state and O is an oracle marking states where |ζ(s)| is close to zero. 2.4 Quantum Entropy Calculation Develop a quantum algorithm to compute H(N) efficiently: 1. Prepare a superposition of prime states up to N 2. Apply a trinary quantum circuit to compute the entropy contribution of each prime 3. Use quantum phase estimation to extract H(N) 2.5 Quantum Information Flow Analysis 2.5.1 Trinary Quantum Walk: Implement a quantum walk on the complex plane to study the behavior of Î(s): U_walk = C ⊗ S Where C is the coin operator and S is the shift operator in trinary space. 2.5.2 Quantum Tensor Network for ζ(s): Represent ζ(s) as a quantum tensor network: ζ(s) ≈ ⟨ψ_L|T(s)^N|ψ_R⟩ Where T(s) is a transfer matrix and |ψ_L⟩, |ψ_R⟩ are boundary states. 2.6 Information-Theoretic Conjectures in Quantum Form 2.6.1 Quantum Entropy Symmetry: Formulate the entropy symmetry conjecture in terms of quantum states: |H(1/2 + it)⟩ = |H(1/2 - it)⟩ 2.6.2 Quantum Maximum Entropy Principle: Express the maximum entropy principle using a quantum variational principle: δ⟨ψ(s)|Ĥ|ψ(s)⟩ = 0 for Re(s) = 1/2 Where Ĥ is a suitable quantum entropy operator. 2.7 Quantum-Enhanced Numerical Verification 2.7.1 Parallel Zero Computation: Use quantum parallelism to check multiple potential zeros simultaneously: |ψ⟩ = (1/√N) ∑ᵢ |sᵢ⟩|ζ(sᵢ)⟩ 2.7.2 Quantum Amplitude Estimation: Apply quantum amplitude estimation to obtain precise values of |ζ(s)| near suspected zeros. 2.8 Quantum Information Potential Implement a quantum circuit to compute the information potential Φ(s): ∇²Φ(s) = -2πÎ(s) Study the behavior of Φ(s) near the critical line using quantum simulation techniques. 2.9 Trinary Quantum Machine Learning 2.9.1 Quantum Neural Network: Train a trinary quantum neural network to recognize patterns in the behavior of ζ(s): |ψ_out⟩ = U_L(...U_2(U_1|ψ_in⟩)) Where Uᵢ are parameterized quantum gates. 2.9.2 Quantum Reinforcement Learning: Use quantum reinforcement learning to optimize search strategies for zeros of ζ(s) in the complex plane. This detailed application combines the power of trinary quantum computing with the insights from the information-theoretic approach to the Riemann Hypothesis. By leveraging quantum superposition, entanglement, and trinary logic, we aim to explore the behavior of ζ(s) and related quantities more efficiently and from novel perspectives. In our next section, we'll analyze the potential advantages and challenges of this combined approach compared to classical and binary quantum methods.

[@ready1fire1aim1]

3. Analysis of Advantages and Challenges 3.1 Potential Advantages 3.1.1 Enhanced Information Density - Advantage: Each qutrit encodes log₂(3) ≈ 1.58 bits, compared to 1 bit per qubit. - Impact: Represent complex values of ζ(s) more compactly, potentially allowing exploration of larger domains. - Comparison: A 100-qutrit system can represent 3¹⁰⁰ ≈ 5 × 10⁴⁷ states, versus 2¹⁰⁰ ≈ 1.27 × 10³⁰ for 100 qubits. 3.1.2 Natural Representation of Complex Plane Regions - Advantage: Three-state system aligns well with the critical strip and regions on either side. - Impact: More intuitive encoding of the complex plane, potentially leading to more efficient algorithms. - Example: Represent the critical strip with a single qutrit state |1⟩, simplifying certain computations. 3.1.3 Quantum Parallelism for Zero-Finding - Advantage: Explore multiple potential zeros simultaneously using superposition. - Impact: Potentially faster verification of the Riemann Hypothesis for larger height T. - Hypothesis: Achieve O(√T) speedup over classical methods for checking zeros up to height T. 3.1.4 Trinary Quantum Fourier Transform - Advantage: More efficient spectral analysis of ζ(s) behavior. - Impact: Better detection of periodicities and patterns in the distribution of zeros. - Possibility: Uncover new connections between the zeros and prime number distribution. 3.1.5 Quantum Information Flow Analysis - Advantage: Study the behavior of I(s) = log|ζ(s)| using quantum simulation. - Impact: Gain new insights into the information-theoretic properties of the zeta function. - Potential: Discover new invariants or conservation laws related to information flow in the complex plane. 3.2 Challenges and Limitations 3.2.1 Hardware Implementation - Challenge: Developing stable, low-error qutrit systems is technically demanding. - Impact: May delay practical implementation of large-scale quantum algorithms for ζ(s). - Mitigation: Focus on hybrid qubit-qutrit systems or specialized qutrit hardware for specific zeta function operations. 3.2.2 Continuous Nature of s - Challenge: Mapping the continuous complex plane to discrete quantum states. - Impact: Potential loss of precision in representing arbitrary complex numbers. - Mitigation: Develop adaptive encoding schemes that focus computational resources on regions of interest. 3.2.3 Quantum Error Correction for Qutrits - Challenge: Error correction for qutrit systems is less developed than for qubits. - Impact: May limit the size and duration of quantum computations involving ζ(s). - Mitigation: Develop new quantum error correction codes optimized for trinary Riemann Hypothesis algorithms. 3.2.4 Classical Simulation and Verification - Challenge: Classical simulation of qutrit systems for algorithm development and verification is more complex. - Impact: May slow down the development and testing cycle. - Mitigation: Develop efficient classical simulators for small qutrit systems focused on zeta function behavior. 3.3 Comparative Analysis 3.3.1 vs. Classical Methods - Advantage: Quantum parallelism allows exploration of multiple s values simultaneously. - Challenge: Achieving quantum advantage requires overcoming significant technical hurdles. - Potential: Verify the Riemann Hypothesis for much larger heights T than classically feasible. 3.3.2 vs. Binary Quantum Methods - Advantage: More compact representation of complex numbers and zeta function values. - Challenge: Less developed hardware and algorithm ecosystem compared to binary quantum computing. - Trade-off: Potential for more efficient zeta function analysis at the cost of increased system complexity. 3.4 Quantitative Estimates (Theoretical) 3.4.1 Algorithmic Speedup - Estimate: For verifying RH up to height T, trinary quantum approach might achieve O(T^(1/3)) time complexity, compared to O(T^(1/2)) for binary quantum and O(T) for classical. - Caveat: Actual speedup heavily dependent on specific algorithms and hardware capabilities. 3.4.2 Precision Requirements - Estimate: Representing ζ(s) to n decimal places requires approximately 0.63n qutrits, compared to n qubits. - Impact: Potentially achieve higher precision with fewer quantum resources. 3.4.3 Entanglement Capacity - Hypothesis: Trinary systems may offer richer entanglement structures relevant to the global behavior of ζ(s). - Exploration: Investigate if trinary entanglement measures reveal new patterns in zeta function zeros. 3.5 Hybrid Approaches 3.5.1 Quantum-Classical Hybrid Algorithms - Concept: Use trinary quantum systems for computationally intensive parts of zeta function analysis, combined with classical post-processing. - Potential: Leverage quantum advantages while utilizing classical strengths in number theory and analysis. 3.5.2 Qutrit-Qubit Hybrid Systems - Approach: Use qutrits for representing complex numbers and qubits for control operations. - Benefit: Combine the information density of qutrits with the more developed qubit technologies. This analysis highlights the significant potential of our trinary quantum information-theoretic approach to the Riemann Hypothesis, while also acknowledging the substantial challenges that need to be overcome. The unique properties of qutrit systems, combined with information-theoretic insights, offer exciting possibilities for advancing our understanding of the zeta function and potentially making progress towards proving the Riemann Hypothesis. In our next section, we'll explore specific aspects of the Riemann Hypothesis where our approach could provide significant insights or computational advantages.

[@ready1fire1aim1]

4. Key Aspects of the Riemann Hypothesis for Our Approach 4.1 Zero Detection and Verification 4.1.1 Trinary Quantum Search for Zeros - Implement a trinary version of Grover's algorithm to search for zeros of ζ(s). - Potential advantage: O(√N) speedup in searching N potential zero locations. - Novel approach: Use quantum walks on a trinary graph representing the complex plane to find zeros. 4.1.2 High-Precision Zero Localization - Employ trinary quantum phase estimation to pinpoint zeros with high accuracy. - Advantage: Represent complex numbers more efficiently using qutrits. - Goal: Verify the Riemann Hypothesis for unprecedented heights on the critical line. 4.2 Analysis of the Critical Strip 4.2.1 Quantum Simulation of ζ(s) in the Critical Strip - Create a quantum circuit that approximates ζ(s) for 0 ≤ Re(s) ≤ 1. - Use trinary superposition to explore multiple s values simultaneously. - Investigate: Quantum signatures of the hypothesized symmetry around Re(s) = 1/2. 4.2.2 Information Flow Visualization - Implement a quantum algorithm to visualize I(s) = log|ζ(s)| across the critical strip. - Leverage trinary quantum states to represent different "levels" of information flow. - Seek: Patterns or structures in the information landscape that might indicate why Re(s) = 1/2 is special. 4.3 Connections to Prime Numbers 4.3.1 Quantum Computation of π(x) - Develop a trinary quantum algorithm to estimate the prime counting function π(x). - Exploit the connection between ζ(s) and prime distribution. - Goal: Achieve quantum speedup in prime counting for large x. 4.3.2 Entropy of Prime Distributions - Implement the quantum version of H(N) = -Σ (p/N) log(p/N) efficiently. - Use trinary quantum circuits to compute entropy for much larger N than classically feasible. - Investigate: How H(N) behaves near values of N related to zeta zeros. 4.4 Functional Equation and Symmetry 4.4.1 Quantum Verification of the Functional Equation - Create a quantum circuit to check ζ(1-s) = 2(2π)^(-s) cos(πs/2) Γ(s) ζ(s). - Utilize trinary encoding to represent the complex components more efficiently. - Explore: Quantum states that inherently respect this symmetry. 4.4.2 Symmetry Detection in Quantum States - Develop quantum algorithms to detect symmetries in the behavior of ζ(s). - Leverage trinary quantum operations that might reveal three-fold symmetries not apparent in binary systems. - Hypothesis: Uncover new symmetries related to the trinary structure of our quantum representation. 4.5 Riemann-Siegel Formula and Computation 4.5.1 Quantum Implementation of Riemann-Siegel Formula - Create a trinary quantum circuit for the Riemann-Siegel formula. - Potential advantage: More efficient computation of ζ(1/2 + it) for large t. - Investigate: Quantum interference effects that might yield insights into the formula's structure. 4.5.2 Error Analysis in Quantum Computations - Develop trinary quantum error correction tailored for zeta function computations. - Study how errors propagate in quantum zeta function evaluations. - Goal: Establish rigorous error bounds for quantum computations of ζ(s). 4.6 Connections to Random Matrix Theory 4.6.1 Quantum Simulation of Random Matrices - Use trinary quantum systems to simulate ensembles of random matrices. - Investigate connections between quantum chaotic systems and zeta function zeros. - Explore: Whether trinary quantum systems reveal new universality classes relevant to ζ(s). 4.6.2 Quantum Computation of Nearest-Neighbor Spacing - Implement quantum algorithms to analyze the spacing between zeta zeros. - Leverage trinary quantum parallelism to compute statistics for many zeros simultaneously. - Compare: Quantum-computed statistics with predictions from random matrix theory. 4.7 Explicit Formulae and Quantum Fourier Analysis 4.7.1 Trinary Quantum Fourier Transform for Explicit Formulae - Develop a trinary QFT tailored for the explicit formulae relating prime powers to zeta zeros. - Potential advantage: More efficient computation of sums over zeros. - Investigate: Whether the trinary QFT reveals new patterns in these relationships. 4.7.2 Quantum Analysis of Primes in Arithmetic Progressions - Create quantum algorithms to study primes in arithmetic progressions using characters and L-functions. - Utilize trinary quantum systems to represent the additional structure of these generalized zeta functions. - Goal: Extend RH investigations to Dirichlet L-functions efficiently. By focusing our trinary quantum information-theoretic approach on these specific aspects of the Riemann Hypothesis, we aim to leverage the unique advantages of our method where they can have the most significant impact. The combination of increased information density, quantum parallelism, and novel information-theoretic perspectives offers the potential for both computational advancements and new theoretical insights. In our next section, we'll discuss the broader implications and potential impact of this approach on number theory and related fields.

[@ready1fire1aim1]

5. Broader Implications and Potential Impact 5.1 Number Theory Revolution 5.1.1 New Computational Paradigm - Potential to revolutionize computational number theory with quantum-enhanced algorithms. - Impact: Solve previously intractable problems in prime number theory and arithmetic progressions. 5.1.2 Theoretical Insights - Information-theoretic perspective may reveal deep connections between primes, entropy, and complex analysis. - Possibility: Uncover new fundamental laws governing prime number distribution. 5.2 Cryptography and Information Security 5.2.1 Post-Quantum Cryptography - Insights into prime number behavior could inform the development of new quantum-resistant cryptographic schemes. - Impact: Enhance cybersecurity in the age of quantum computing. 5.2.2 Quantum Random Number Generation - Utilize the apparent randomness of zeta zeros for high-quality quantum random number generation. - Application: Improve cryptographic key generation and Monte Carlo simulations. 5.3 Quantum Computing Advancement 5.3.1 Trinary Quantum Architecture Development - Drive innovation in qutrit-based quantum hardware to support RH investigations. - Spillover: Advancements in qutrit technology could benefit quantum computing in general. 5.3.2 New Quantum Algorithms - Algorithms developed for RH could find applications in other areas of quantum computation. - Potential: Inspire new approaches to quantum simulation and optimization problems. 5.4 Complex Systems and Chaos Theory 5.4.1 Quantum Chaos Connections - Deepen understanding of links between quantum chaotic systems and the Riemann zeta function. - Implication: New insights into the quantum-classical transition in chaotic systems. 5.4.2 Universality in Complex Systems - Explore whether the universality observed in zeta zeros extends to other complex systems. - Impact: New universal laws governing seemingly unrelated complex phenomena. 5.5 Theoretical Physics 5.5.1 Quantum Field Theory - Investigate connections between zeta function regularization in QFT and our quantum approach to RH. - Potential: New methods for handling infinities in quantum field theories. 5.5.2 Quantum Gravity - Explore links between the distribution of primes and fundamental theories of quantum gravity. - Speculation: Insights into the structure of spacetime at the Planck scale. 5.6 Data Science and Machine Learning 5.6.1 Quantum Machine Learning for Number Theory - Develop quantum ML algorithms for pattern recognition in prime number distributions. - Application: Automated conjecture generation in number theory. 5.6.2 Quantum-Enhanced Data Analysis - Apply insights from zeta function analysis to big data problems. - Potential: New techniques for detecting hidden periodicities in large datasets. 5.7 Computational Complexity Theory 5.7.1 Quantum Complexity Classes - Refine understanding of quantum complexity classes through RH-related problem solving. - Impact: Clarify the power and limitations of quantum computation. 5.7.2 New Complexity Measures - Develop information-theoretic complexity measures inspired by our approach to RH. - Potential: Novel ways to classify the difficulty of mathematical and computational problems. 5.8 Philosophy of Mathematics 5.8.1 Nature of Mathematical Truth - Insights from quantum approaches to RH could inform debates about mathematical platonism vs. formalism. - Question: Does the quantum nature of reality have implications for the foundations of mathematics? 5.8.2 Limits of Knowability - Explore connections between quantum uncertainty and fundamental limits on mathematical knowledge. - Implication: New perspectives on Gödel's incompleteness theorems in a quantum context. 5.9 Interdisciplinary Collaborations 5.9.1 Mathematics-Physics-Computer Science Nexus - Foster deeper collaborations between these fields, centered on RH investigations. - Benefit: Cross-pollination of ideas leading to breakthroughs in multiple disciplines. 5.9.2 Quantum Biology Connections - Investigate whether prime number patterns and zeta function properties have analogues in biological systems. - Speculation: New models for understanding genetic codes and neural networks. 5.10 Educational Impact 5.10.1 Quantum-Enhanced Mathematics Education - Develop educational tools that use quantum visualizations to teach complex mathematical concepts. - Goal: Make advanced number theory more accessible and intuitive to students. 5.10.2 Public Engagement with Mathematics - Use the excitement around quantum approaches to RH to increase public interest in mathematics. - Outcome: Greater appreciation and funding for fundamental mathematical research. This combined trinary quantum and information-theoretic approach to the Riemann Hypothesis has the potential to not only advance our understanding of this specific problem but also to catalyze progress across a wide range of scientific and mathematical disciplines. By bridging number theory, quantum computing, and information theory, we open up new avenues for discovery and innovation that could reshape our understanding of mathematics, computation, and the fundamental nature of information in our universe. In our final section, we can outline next steps and future research directions for this ambitious program.

[@ready1fire1aim1]

6. Next Steps and Future Research Directions 6.1 Theoretical Foundations 6.1.1 Formalize Trinary Quantum Information Theory - Develop rigorous mathematical framework for trinary quantum states and operations. - Establish theorems linking trinary quantum information to zeta function properties. 6.1.2 Information-Theoretic Zeta Function Axioms - Propose a set of axioms characterizing ζ(s) in terms of information flow and entropy. - Investigate whether these axioms necessarily lead to the Riemann Hypothesis. 6.2 Algorithm Development 6.2.1 Trinary Quantum Zeta Evaluation Algorithm - Design and optimize a quantum circuit for computing ζ(s) using qutrits. - Benchmark against classical and binary quantum algorithms. 6.2.2 Quantum Zero-Finding Heuristics - Develop quantum-inspired classical algorithms for locating zeta zeros. - Create hybrid quantum-classical algorithms for efficient zero verification. 6.3 Quantum Hardware Collaboration 6.3.1 Qutrit Quantum Processor Design - Collaborate with quantum hardware teams to develop qutrit-based processors. - Focus on architectures optimized for zeta function computations. 6.3.2 Error Mitigation Techniques - Develop error correction and mitigation strategies tailored for RH-related quantum computations. - Investigate topological quantum computing approaches for robust zeta function simulation. 6.4 Numerical Experiments and Data Analysis 6.4.1 Large-Scale Quantum Simulations - Conduct extensive numerical experiments using quantum simulators and available quantum hardware. - Analyze patterns in quantum states representing ζ(s) for large heights. 6.4.2 Machine Learning on Quantum Data - Apply classical and quantum machine learning to the data generated from quantum zeta function simulations. - Seek novel patterns or structures that might suggest paths towards a proof. 6.5 Interdisciplinary Connections 6.5.1 Quantum Chaos and RH - Deepen investigations into connections between quantum chaotic systems and zeta zeros. - Explore whether trinary quantum systems reveal new universality classes relevant to RH. 6.5.2 Quantum Thermodynamics of Primes - Develop a thermodynamic theory of prime numbers based on quantum information principles. - Investigate phase transitions in prime number behavior using quantum statistical mechanics. 6.6 Proof Strategies 6.6.1 Quantum Information Conservation Laws - Formulate and prove conservation laws for quantum information flow in ζ(s). - Investigate whether these laws constrain the locations of zeta zeros. 6.6.2 Topological Quantum Approaches - Explore topological quantum computation models for representing ζ(s). - Seek topological invariants that might necessitate the truth of RH. 6.7 Computational Complexity Analysis 6.7.1 Quantum Complexity of RH - Analyze the quantum computational complexity of RH-related problems. - Investigate whether RH can be reformulated as a problem in BQP (Bounded-error Quantum Polynomial time). 6.7.2 Quantum-Classical Complexity Gaps - Identify specific RH-related tasks where quantum approaches offer provable speedups. - Develop a hierarchy of RH sub-problems based on their quantum vs. classical complexity. 6.8 Educational and Outreach Initiatives 6.8.1 Quantum RH Visualization Tools - Develop interactive quantum simulations for visualizing zeta function behavior. - Create educational materials linking quantum computing, information theory, and number theory. 6.8.2 Collaborative Research Platform - Establish an online platform for researchers to share quantum RH algorithms and results. - Organize regular workshops and conferences on quantum approaches to RH. 6.9 Ethical and Philosophical Considerations 6.9.1 Implications for Mathematical Truth - Engage philosophers of mathematics in discussions about the meaning of quantum approaches to classical conjectures. - Explore the epistemological status of quantum-assisted mathematical discoveries. 6.9.2 Responsible Innovation - Consider the potential impacts of RH resolution on cryptography and cybersecurity. - Develop guidelines for responsible use of quantum technologies in mathematical research. 6.10 Long-term Vision 6.10.1 Quantum Institute for Mathematical Physics - Propose the establishment of a dedicated research institute combining quantum computing, information theory, and mathematical physics. - Foster long-term, high-risk research aimed at fundamental mathematical problems. 6.10.2 Generalized Quantum Approaches to Open Problems - Extend the methodologies developed for RH to other open problems in mathematics. - Aim to create a new paradigm of quantum-assisted mathematical research. This roadmap provides a comprehensive guide for advancing our trinary quantum information-theoretic approach to the Riemann Hypothesis. It emphasizes the need for progress on multiple fronts: theoretical development, algorithmic innovation, hardware advancement, and interdisciplinary collaboration. The journey ahead is challenging but immensely exciting. Even if this approach doesn't immediately lead to a proof of the Riemann Hypothesis, the insights gained and methods developed could revolutionize our understanding of number theory, quantum computation, and the deep connections between mathematics and physics. By pursuing this ambitious program, we not only strive to resolve one of the greatest open problems in mathematics but also push the boundaries of quantum computing and information theory, potentially ushering in a new era of mathematical discovery and computational capability.

[@pandabearguy1]

My favorite is the generalized stokes theorem

[@journeymantraveller3338]

Very clear discussion. Great!

[@g0rgth3b0rg]

Very clear explanations. Thank you for the great content!

[@EricLeePiano]

your passion and enthusiasm is infectious! you are one of the best math teachers on youtube. 😊

[@nedmerrill5705]

_Prime Obsession,_ by John Derbyshire.

[@Ninja20704]

Here is one interesting result you can show using this formula. If you were to choose s positive integers at random, the probability they all are collectively coprime is 1/zeta(s) The probability a prime p divides a random positive integer is 1/p. The probability all the numbers do is 1/p^s . Thus, the probability that the prime doesn’t divide all the numbers at the same time is 1-1/p^s. If the numbers are coprime, that means no prime number divides all of them. So to get that probability, we just multiply (1-1/p^s) over all prime numbers p. This is precisely the reciprocal of the product shown in the formula, which is why the result is just the reciprocal of the zeta function.

[@DrTrefor]

Nice ones!

[@Ninja20704]

⁠@@DrTrefor also another way to prove the result I’ve seen is that on the RHS, each 1/(1-1/p^s) is actually a geometric series. So you get for example 1/(1-1/2^s) = 1 + 1/2^s + 1/4^s + 1/8^s + … with all the powers of that prime. So when you multiply and distribute all those brackets out, you will get all positive integers exactly once due to fund thm of arithmetic since every possible prime factorisation will appear exactly once somewhere

[@kaifscarbrow]

I've been following YT math sphere for over 10 years now and this is the FIRST time I've come across this formula. And for anybody who's saying that euler's identity is the most beautiful equation, YOU ARE DELUSIONAL!! Look at this! It has addition equal to multiplication WITHOUT Log!! And there's all the primes and all the natural numbers too!!!

[@jasimmathsandphysics]

I remember getting that equation and I was very happy to see the relationship

[@jedediahjehoshaphat]

I suppose our proficiencies and exposure are big factors in one's personal assessment of beauty in mathematical equations. Me personally, i prefer the Navier Strokes equation; it basically generalises diffusion and transport phenomena of liquid flow, making it an equation that governs the whole world ( approximately)

[@DrTrefor]

Don’t get me wrong, that should ALSO be on most shortlists:D

[@mcpecommander5327]

Also Maxwell’s equation (∇ F=J) is very nice

[@bjornfeuerbacher5514]

The Navier-Stoles equation and Maxwell's equation are two of the most beautiful equations of physics - not of mathematics. ;)

[@elltwo8393]

Riemann-Roch and Gauss-Bonnet would be my pick from your list. More generally the index theorem I find beautiful.

[@ratandmonkey2982]

what is 1/p + 1/q + 1/r = 1 ?

[@RSLT]

I 1000000% agree. This is the most beautiful bridge, connecting prime numbers, which are unpredictable, to a well-ordered, very predictable set of natural numbers (n+1). The hair on the back of my neck stands up whenever I see it. Great video-I love it so much!"❤🧡💛💚💙💜🤎🖤🤍

[@bjornfeuerbacher5514]

What is the (n+1) in your comment supposed to mean?

[@RSLT]

@bjornfeuerbacher5514 The members of the set of prime numbers are unpredictable. However, for natural numbers, you can find the next element in the set by adding one to the previous member.

[@evid-rz3nu]

amazing n very very nice thanks a lot for this video. love from India

[@DrTrefor]

Glad you enjoyed!

[@theflaggeddragon9472]

Do Riemann-Roch next!! 😈

[@DrTrefor]

I haven't been brave enough yet to do it justice:D

[@imeprezime1285]

There are many beautiful formulae in math. The opinions of their aesthetical appeal and importance differ greatly. I'd say that Pythagorean theorem and its expression is the most important one

[@abublahinocuckbloho4539]

10:06 explain how the negative sign in front of the sum disappears from the expanded form the sum of reciprocal powers of primes at the bottom of the screen

[@DrTrefor]

When taking logs, the negatives out front become exponents.

[@xyz.ijk.]

My favorite are the gamma function and W Lambert

[@DrTrefor]

Nice ones!

[@graf_paper]

What? Is this the kind of content we are going to be getting from you while on Sabbatical? 🎉🎉🎉 This was an excellent presentation, so looking forward to what comes next from this channel.

[@DrTrefor]

Lots more coming!!

[@stvp68]

Cool proof!

[@Npvsp]

I believe that first of all one should specify and differentiate between equation, function, formula and identity. Using “equation” a gogo is wrong. For example: Euler’s identity. NOT and equation, it’s an identity. Et cetera.

[@juniorcyans2988]

Fascinating!❤❤❤

[@markrockliff2742]

There seems to be a theme of quantum computation running throughout this video. 11:49 / 17:03 was this really simply a convergence of three laws? Law: 3, law: 1, and Law: 7? with the unification on the left-hand side and the helix on the right-hand side. Infinity being the multiplicity and n=1 the unity?

[@spicemasterii6775]

Your shirt is awesome

[@DrTrefor]

Haha I love it so much

[@hedu5303]

Wow this guy is a god in math

[@uselesscommon7761]

My favorite equation is the formula for the probably of N successes in independent M trials of P success probability. It's a seemingly ugly mess with a beautifully elegant meaning behind every symbol

[@DrTrefor]

Nice one!

[@maaikevreugdemaker9210]

I can't find the 3b1b link in the description. Thanks a lot for explaining this beautifull equation and how it was derived.

[@albireo151]

kzbin.info/www/bejne/qXWTf52YrNafj9k

[@ericmarseille2]

5:00 I'm a quiche at math but shouldn't 1/2s times zeta be more like 1/2s + 1/4s² + 1/6s²... etc.?

[@joserimada1331]

navier stokes equation diffusion term is missing the velocity "u"

[@DrTrefor]

Dope!

[@andrewharrison8436]

17 minutes watching the video, 10 minutes on the comments. Still a finite amount of time. Following up analytic continuation, rewatching to see why the expansion of the products doesn't double count some terms then any further questions raised - hmm I don't see this as a finite sum. What the heck, it's all fun.

[@louisreinitz5642]

The log graph on your t-shirt is wrong.

[@repbacchista]

could someone please name the other formulas that are showed at 0:06?

[@bjornfeuerbacher5514]

See the comment by John Cessant.

[@ivornworrell]

my fav math er is anything that has real life applications, thx

[@milos_radovanovic]

that shirt is wrong for the y=log x

[@bjornfeuerbacher5514]

It's right if the base a of the logarithm is a number between 0 and 1.

[@DoxxTheMathGeek]

Okay, this is my second favorite proof now I suppose, it is awesome. X3 Thank you so much for showing! >w

[@DrTrefor]

I'm glad you like it!

[@INT41O]

The y=log_a x stick figure......... 🙃

[@sapwho]

❤️❤️

[@familychannel3826]

Hello, I have discovered mathematical formulas that can identify all non-prime numbers, allowing me to deduce all prime numbers. Does this discovery hold any scientific value?

[@Chiavaccio]

👏👏👍

[@roshan0405]

❤

[@schizoframia4874]

What the sigma😮

[@xBassthh]

Where can I buy that shirt? Also, nice video. 👌

[@DrTrefor]

My wife gave it to me, I don’t know!

[@mvignesh7705]

Although limit for 1/x as x tends to infinity is 0, why does integral over it from x=c>0 to infinity diverge?

[@DrTrefor]

It isn't "intuitively" obvious which way it should work. You are adding more of a quantity getting smaller, it is a debate about whether it gets small enough fast enough to converge or not. It turns out that for 1/x^s the behaviour of converge/diverge changes at s=1.

[@loopingdope]

The integral is bigger than sum 1/n

[@mvignesh7705]

@@DrTrefor got it sir, extremely small doesn't imply adding more of it limit that to a particular value, Thanks a lot for a response, sir

[@mvignesh7705]

@@loopingdope Thanks for another kinda perspective, saw a mathematical formulation of this divergence exponentially slower, here's the intuition behind that explained by @DrTrefor kzbin.info/www/bejne/a5bNnpqtjbuGp7Msi=AdpgZMxYNFbkUiRH

[@dantevaldez2325]

Ngl I was staring at that shirt the whole time thinking where can I get that…

[@user-rn9bu3ce6l]

拼多多上应当有这件衣服

[@davidespinosa1910]

Math is about structure, not equations.

[@guidomista8448]

Hey, this isnt to the equation in the video, but you're shirt is wrong, it says that Log base a of x is a downwards slope, which we know is wrong. Should've put a negative sign out infront, or make it a^-x.

[@GrifGrey]

maybe the camera is mirrored?

[@GrifGrey]

nevermind, it is just that one. funny

[@Ninja20704]

You can have 0

[@erfanmohagheghian707]

A million dollars is too little to be tempting to work on such a hard-to-prove or disprove hypothesis. You can barely buy a good house with it. I guess you make more from advertising Brilliant.

[@DrTrefor]

Ha I wish!

[@erfanmohagheghian707]

@@DrTrefor 😊 You're a good knowledgeable prof. I wish we could collaborate. Let me know if you happened to come for a visit to Vancouver

[@tommushrom5929]

In your dreams. E=mc2.

[@binbots]

Math is just counting an infinite amount of zeros.

[@comdo777]

asnwer= 2/5 or 2 🤣

[@comdo777]

my like asnwer=1 isit what compone nt 1/2 what math matter isit

[@bjornfeuerbacher5514]

Say, do you like making a public fool of yourself? Or why do you always write totally nonsensical answers below lots of math videos?

[@realzeelink]

So many people are probably wondering why after watching this video they are not earning any money automagically while most drug dealers and so called self proclaimed law enforcers are getting on top without the knowledge of mathematics.

[@user-gj6cw6yc8s]

😊 revival It should be the only key word the human being knew I promise

[@user-gj6cw6yc8s]

😊 it's Doomsday