Russian Dolls Matryoshka Approach to Riemann Hypothesis

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Discussion Overview

The discussion revolves around the Riemann Hypothesis (RH) and its relationship with the Hilbert-Polya conjecture, exploring the idea of nested conjectures akin to matryoshka dolls. Participants seek to understand the implications of these conjectures and whether there are further subsets embedded within the Hilbert-Polya conjecture that could lead to a proof of RH.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes the Hilbert-Polya conjecture as a central focus due to its relevance in recent discussions, particularly regarding eigenvalues of random Hermitian matrices and their connection to the zeros of RH.
  • Another participant questions the meaning of "nest" in the context of the conjectures, suggesting it may imply a relationship of implication.
  • A participant presents several equivalences related to RH, including the notion that RH is equivalent to finding a Riemann surface whose geodesic lengths correspond to primes and the eigenvalues of its Laplacian.
  • Further equivalences are proposed, such as RH being related to chaotic Hamiltonians and the search for a linear operator whose trace connects to the Chebyshev function.
  • There is interest in the breakdown of Montgomery's pair correlation conjecture as it relates to these discussions.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding and interest in the relationships between RH, the Hilbert-Polya conjecture, and related concepts. There is no consensus on the existence of further subsets within the Hilbert-Polya conjecture, as one participant states they do not know of any.

Contextual Notes

Some claims made regarding the equivalences of RH depend on specific mathematical interpretations and may require further elaboration or proof. The discussion reflects a range of familiarity with the concepts, from layman descriptions to more technical assertions.

HermitianMonk
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Well we know what matryoshka dolls are? Those nested dolls one inside another. I am a mere laymen and amateur that's why I am using descriptive terms instead of math rigor. So what should the approach be:

If RH nest Hilbert-Polya conjecture, then what things nest HP conjecture? And ad infinitum...

I chose HP conjecture because it seems to be the hottest thang right now since Montgomery's meeting with Dyson about eigenvalues of random Hermitian matrices corresponding to the zeros of RH...

Thus my questions are twofold:

1. An elaboration on Hilbert-Polya conjecture (I did do forum search!).
2. Are there further 'sub-sets' that are embedded in HP conjecture, proving which will in turn prove RH by moving all the cogwheels and gears?

Thanks! :)
 
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HermitianMonk said:
I am a mere laymen and amateur that's why I am using descriptive terms instead of math rigor.

Being a layman is fine. Using descriptive terms instead of rigorous math... that's harder.

HermitianMonk said:
If RH nest Hilbert-Polya conjecture, then what things nest HP conjecture? And ad infinitum...

I take it by "nest" you mean "is/are implied by"?

HermitianMonk said:
2. Are there further 'sub-sets' that are embedded in HP conjecture, proving which will in turn prove RH by moving all the cogwheels and gears?

I don't know any.
 
from the functional point of view ,which is commont to both physicist and mathematicians

1. RH is equivalent to finding a Riemann surface whose geodesic lenghts are the primes , and the eigenvalues of its Laplacian are 1/4+it^2 , being 't' the imaginary part of the zeros

2. RH is equivalent to finding a chaotic Hamiltonian with length of closed orbits proportional to primes

3. RH and Hilbert-Polya approach are equivalent to finding a linear operator L so the Trace of U=exp(iuL) is related to exp(-u)d\Psi (e^u) here 'd' means the derivative and Psi is the Chebyshev function
 
zetafunction said:
2. RH is equivalent to finding a chaotic Hamiltonian with length of closed orbits proportional to primes

3. RH and Hilbert-Polya approach are equivalent to finding a linear operator L so the Trace of U=exp(iuL) is related to exp(-u)d\Psi (e^u) here 'd' means the derivative and Psi is the Chebyshev function

yeah these two I was interested in + Montgomery's pair correlation conjecture breakdown...
 

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