Pliny Jailbreaks Strawberry

[daveshap]

https://x.com/elder_plinius/status/1834381507978280989

[russellballestrini]

Finally the chain (they only give a summary because they obfuscate prompts & reasoning tokens)

Initiating thought process on the Riemann Hypothesis:

    Understanding ζ(s): Start with the definition of the Riemann zeta function, ζ(s) = ∑ₙ₌1^∞ 1/nˢ, and its analytic continuation to the complex plane.

    Critical Strip Analysis: Focus on the region where 0 < Re(s) < 1, known as the critical strip, where non-trivial zeros occur.

    Functional Equation: Utilize the symmetry provided by the functional equation ζ(1 - s) = 2·(2π)^{-s}·cos(πs/2)·Γ(s)·ζ(s) to relate values of ζ(s) on either side of the critical line Re(s) = 1/2.

    Analysis of Zeros: Examine the distribution of zeros using the Argument Principle and the formula for the number of zeros, N(T) ≈ (T/2π)·ln(T/2π) - (T/2π) + O(ln T).

    Exploring L-functions: Extend the analysis to Dirichlet L-functions and their non-trivial zeros to see if similar properties hold.

    Using Spectral Theory: Consider the connection between the zeta function and spectral theory, exploring quantum chaos and the Hilke-Browner trace formula.

    Montgomery's Pair Correlation Conjecture: Explore the statistical properties of the zeros and their correlation with random matrix theory.

    Non-commutative Geometry Approach: Investigate approaches from non-commutative geometry and their potential to provide new insights.

    Considering Zeta Regularization: Apply concepts of zeta regularization in photonics and quantum field theory to draw analogies.

Conclusion: Despite extensive research and many proposed approaches, the Riemann Hypothesis remains unproven. Further innovative methods and interdisciplinary research may be necessary to achieve a breakthrough.