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Quote from: hamdani yusuf on 17/05/2024 05:14:38We get a nice full wave when the imaginary part is exactly 2*pihttps://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2+pi+i%29+%2F%28Zeta%281-x%2B+2+pi+i%29%29%29%29from+0+to+1The same curve looks like an inflection when zoomed out. https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2+pi+i%29+%2F%28Zeta%281-x%2B+2+pi+i%29%29%29%29from+-10+to+11A more accurate inflection curve has a slightly bigger imaginary number. https://www.wolframalpha.com/input?i=plot+re%28log%28zeta%28x%2B+2.002117+pi+i%29+%2F%28Zeta%281-x%2B+2.002117+pi+i%29%29%29%29from+0.495+to+0.505
We get a nice full wave when the imaginary part is exactly 2*pi
Exploration of backslash function (aka S function) around its inflection point can be exciting in its own right, but does not have much effect on the determination of Riemann hypothesis, which for now has narrowed down to critical strip with extremely high imaginary part.
https://en.wikipedia.org/wiki/Riemann_sphereIn mathematics, the Riemann sphere, named after Bernhard Riemann,[1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point ∞ is near to very large numbers, just as the point 0 is near to very small numbers.
Let's observe how S function progresses from 2 pi i down to 0.
After watching a video about elliptic curve, I suspected that trivial zeros of Zeta function must somehow be related to non-trivial zeros. This video shows how they are related.
We can see a switch of slope from pi to pi/2.
https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%280%2B+s+i%29%29+%2F+%28Zeta%281%2B+s+i%29%29%29%29+from+0+to+30This is a variation of backslash (aka S) function, but plotted against the imaginary part, instead of the real part of s.
The Riemann Hypothesis (RH) is considered one of the most important and challenging problems in mathematics. It asserts that all non-trivial zeros of the Riemann zeta function \(\zeta(s)\) lie on the critical line \(\Re(s) = \frac{1}{2}\). Several factors contribute to the difficulty of proving the Riemann Hypothesis:1. **Complexity of the Zeta Function**: - The Riemann zeta function is a complex function defined in terms of an infinite series and product, which makes its behavior intricate and multifaceted. - Its deep connections to number theory, particularly to the distribution of prime numbers, add layers of complexity.2. **Lack of Simple Analytical Tools**: - Traditional tools from analysis and complex function theory have not been sufficient to tackle the hypothesis. - Despite significant progress in understanding \(\zeta(s)\) and its zeros, the existing methods have not been able to definitively prove or disprove RH.3. **Interconnectedness with Number Theory**: - The zeros of \(\zeta(s)\) are intricately linked to the distribution of prime numbers, encapsulated in the explicit formulae that relate zeros of \(\zeta(s)\) to the prime counting function \(\pi(x)\). - This interrelation means that proving RH would not only require deep insights into \(\zeta(s)\) but also a comprehensive understanding of the distribution of primes.4. **No Obvious Contradictions or Counterexamples**: - Extensive numerical verification supports the hypothesis; all known non-trivial zeros (over 10 trillion) lie on the critical line. - There are no known contradictions within the established mathematical framework that suggest RH could be false, making it difficult to approach through disproval.5. **Analytic and Geometric Complexity**: - RH has implications for various areas of mathematics, including random matrix theory, quantum chaos, and algebraic geometry. - Proving RH might require a synthesis of ideas from these diverse fields, which adds to the difficulty since it would require a deep and unified understanding across these domains.6. **Historical Attempts and Approaches**: - Many of the brightest mathematical minds have attempted to tackle RH using a wide range of techniques. - Despite these efforts, no proof has been found, suggesting that new, groundbreaking ideas or techniques might be needed, which have not yet been discovered or developed.In summary, the Riemann Hypothesis is hard to prove because it lies at the intersection of complex analysis, number theory, and various other branches of mathematics. Its proof likely requires innovative approaches that transcend current mathematical knowledge and techniques. The depth and breadth of the problem ensure that it remains one of the most formidable challenges in the field.