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/sci/ - Science & Math

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16148367 No.16148367 [Reply] [Original]

so, new gpt2 architecture chatbot has been shadowdropped and it's output is significally different from gpt4 in some cases. below is what happens when asked to prove riemann's hypothesis and the overall process is consistent between multiple regens (normal gpt4t proposes a meme solution). can some mathfag look if any of this makes any sense? sorry for broken formulas formatting, that's how gpt2 outputs them (i think you can paste them in latex formula viewer to see them properly):

The Riemann Zeta function, (\zeta(s)), is defined for (\text{Re}(s) > 1) by the absolutely convergent series:

[
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}
]

and extended to the rest of the complex plane (except (s = 1)) via analytic continuation. The functional equation for (\zeta(s)) is central in its analysis:

[
\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)
]

The Riemann Hypothesis asserts that all non-trivial zeros of (\zeta(s)) lie on the critical line (\text{Re}(s) = \frac{1}{2}).

>> No.16148369

>>16148367
Step 1: Quantum Modular Forms
Building on the work in quantum modular forms by Don Zagier and others, suppose there exists a quantum modular form that connects with the Zeta function in a way that impacts the location of its zeros. Define a quantum modular form (K(z)) that exhibits transformation properties not only under the action of the modular group (SL(2, \mathbb{Z})) but also under a quantum group.

Conjecture:
[
K(z) \xrightarrow{z \to -1/z} z^\kappa K(z) \quad \text{(quantum symmetry)}
]
where (\kappa) is a critical exponent potentially related to the zeros of (\zeta(s)).

>> No.16148371

>>16148369
Step 2: Connection to Zeta Zeros
Utilize the newly defined quantum modular form to establish a connection to (\zeta(s)). Specifically, use the Mellin transform which relates decay properties of a function as (x \to 0^+) to the distribution of its poles and zeros.

Mellin Inversion Formula Applied:
[
\int_0^\infty K(x) x^{s} \frac{dx}{x} = \Phi(s) \zeta(s)
]
for some function (\Phi(s)) deeply tied to the quantum modular properties of (K(x)).

>> No.16148373

>>16148371
Step 3: Analytic Properties and the Critical Line
Analyze the analytic properties of (\Phi(s)) to control the zeros of (\zeta(s)). The quantum symmetry and transformation properties under the modular group provide constraints which (\Phi(s)) must satisfy, particularly influencing the argument of (\zeta(s)) along critical lines.

Critical Constraint:
[
\text{Arg}(\Phi(\frac{1}{2} + it)) + \text{Arg}(\zeta(\frac{1}{2} + it)) = 0 \quad \text{(mod } \pi\text{)}
]

Step 4: Verification Through the Argument Principle
Use the Argument Principle in complex analysis to verify that the zeros of (\zeta(s)) induced by (\Phi(s)) must lie on the critical line. The symmetry properties of (K(z)) and its relation through (\Phi(s)) ensure that any deviation from the critical line leads to a contradiction in the modular and quantum symmetries.

Step 5: Completion and Implications
Having established these properties, complete the proof by rigorously showing through contour integration and the functional equation that the only possible locations for zeros of (\zeta(s)) that satisfy all given conditions are on the critical line (\text{Re}(s) = \frac{1}{2}).

>> No.16148398

>>16148367
make better threads