Request for Assistance with E8 Automorphic L-function Computation

[Hector Luis Avila Pisetta]

Dear LMFDB Team,

I am working on a project exploring the connection between twin primes and the E8 Lie group via the Langlands program. We’ve developed a unified model integrating icosahedral encoding, wavelet transform analysis, and approximate Satake parameters for twin primes up to 10^6. Our wavelet analysis reveals non-stationary periodicities in the twin prime gaps with dominant frequencies (0.4603, 0.3420, 0.3147, 0.1236), corresponding to periods (2.17, 2.92, 3.18, 8.09).

We’ve approximated an E8 automorphic L-function by using a genus 2 Siegel modular form as a proxy, but the zero spacing (average 1.4114, frequency 0.7085) doesn’t match our wavelet frequencies. We believe a genus 8 Siegel modular form (related to the A_8 lattice, which connects to E8) or a direct E8 automorphic L-function would provide a more accurate zero distribution.

Could you assist in computing the L-function for a genus 8 Siegel modular form (e.g., on GSp(16, Z), level 1, weight 8) or an E8 automorphic L-function for a small range of primes (e.g., p < 20)? We’re particularly interested in the Satake parameters and the first few non-trivial zeros on the critical strip to compare with our wavelet frequencies.

Thank you for your time and assistance!

https://sagecell.sagemath.org/?q=hbxhug

Best regards,

--

Hector Luis Avila Pisetta
USA.
dula...@gmail.com

[David W. Farmer]

Dear Hector Luis Avila Pisetta,

I am writing only for myself: other LMFDB people may think
differently.

I'd need to hear some details before I could give serious consideration
to a proposed connection between twin primes, E8, and the Langlands
program. Perhaps you have written something which answers these
questions:

1) What are you claiming beyond the well-established heuristics for
twin primes?

2) How does the connection to E8 arise?

3) What does it mean to "approximate" an E8 automorphic L-function?
As far as I know, there is no sense in which two different L-function
can be close together: by Selberg orthogonality, two distinct L-functions
are completely different than each other. Their zeros will have no
relationship to each other.

3) The zero spacings of an L-function are completely understood, arising
in a straightforward way from the parameters in the functional equation.
The actual zeros are just perturbations of a main term. Furthermore,
there is no global average spacing, only a local average. In what sense
it is meaningful to use the handful of zeros in your sage calculation?

4) What, other than numerology, makes you interested in a particular
Siegel modular form L-function? And what is its connection to E8?

5) If there really is a connection to some L-function arising from
E8, which one are you looking for? What is its degree, conductor, etc?

Regards,

David Farmer

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