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PreprintCC BY-NC-ND 4.0HAL

From scalar rigidity to abstract prime set and abstract integer system

  • W Oukil
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Abstract

We propose a unified framework for Prime Rigidity Theory (PR) by introducing the notions of an "abstract integer system" and an "abstract prime set", consisting, respectively, of two unbounded sequences 𝒫 and 𝒩 of elements of 𝒳, where 𝒳 denotes the algebra of bounded linear endomorphisms of a complex Banach space X, such that every element of 𝒩 can be expressed as a finite product of powers of elements of 𝒫. We define the zeta function ΞΆX,𝒩 on π’³βˆ–IX, where IX is the identity endomorphism. The scalar restriction ΞΆβ„‚,𝒩 yields generalized Beurling zeta functions; in particular, ΞΆβ„‚,β„• coincides with the classical Riemann zeta function. We introduce the notions of a "rigid abstract integer system" and "rigid abstract prime set" in order to preserve the rigid asymmetric structure obtained in the scalar case.

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Cite

BibTeXRIS
BibTeX
@misc{oukil2026from,
  title = {{From scalar rigidity to abstract prime set and abstract integer system}},
  author = {W Oukil},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  url = {https://hal.science/hal-05682635v3},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - From scalar rigidity to abstract prime set and abstract integer system
AU  - W Oukil
PY  - 2026
DA  - 2026/07/06
PB  - HAL
UR  - https://hal.science/hal-05682635v3
AB  - We propose a unified framework for Prime Rigidity Theory (PR) by introducing the notions of an "abstract integer system" and an "abstract prime set", consisting, respectively, of two unbounded sequences 𝒫 and 𝒩 of elements of 𝒳, where 𝒳 denotes the algebra of bounded linear endomorphisms of a complex Banach space X, such that every element of 𝒩 can be expressed as a finite product of powers of elements of 𝒫. We define the zeta function ΞΆ_X,𝒩 on π’³βˆ–I_X, where I_X is the identity endomorphism. The scalar restriction ΞΆ_β„‚,𝒩 yields generalized Beurling zeta functions; in particular, ΞΆ_β„‚,β„• coincides with the classical Riemann zeta function. We introduce the notions of a "rigid abstract integer system" and "rigid abstract prime set" in order to preserve the rigid asymmetric structure obtained in the scalar case.
ER  -