Cancellation profiles of Fibonacci-Lucas zeta sums

Abstract

Finite Fibonacci and Lucas sums involving zeta values at negative integers can have individual terms far larger than their total. In this paper, we study the distribution of these term magnitudes as the even degree increases. For non-negative weights of at most exponential growth, we prove a finite exponential expansion for the normalized absolute-term sum with an explicit remainder at every fixed order. The normalized magnitudes converge in total variation to a discrete distribution determined by the even exponential generating function of the weights. For the Fibonacci and Lucas sums, this distribution has its unique maximum at index eight and we prove that the same index gives the largest finite-sum term at every even degree at least twenty-six. We also derive the factorial growth of the summation condition number. The signed identities are placed within the established Bernoulli-polynomial reflection framework and extended to a trace-one recurrence family, including repeated roots and a vanishing scale. Exact rational calculations and precision-refined evaluations illustrate the proved bounds and distinguish a fixed cancellation profile from the growing length of the sum.

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BibTeX RIS

BibTeX

@misc{danesh2026cancellation,
  title = {{Cancellation profiles of Fibonacci-Lucas zeta sums}},
  author = {Payam Danesh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33564},
  eprint = {2609.33564},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33564},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Cancellation profiles of Fibonacci-Lucas zeta sums
AU  - Payam Danesh
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33564
UR  - https://arxiv.org/abs/2609.33564
AB  - Finite Fibonacci and Lucas sums involving zeta values at negative integers can have individual terms far larger than their total. In this paper, we study the distribution of these term magnitudes as the even degree increases. For non-negative weights of at most exponential growth, we prove a finite exponential expansion for the normalized absolute-term sum with an explicit remainder at every fixed order. The normalized magnitudes converge in total variation to a discrete distribution determined by the even exponential generating function of the weights. For the Fibonacci and Lucas sums, this distribution has its unique maximum at index eight and we prove that the same index gives the largest finite-sum term at every even degree at least twenty-six. We also derive the factorial growth of the summation condition number. The signed identities are placed within the established Bernoulli-polynomial reflection framework and extended to a trace-one recurrence family, including repeated roots and a vanishing scale. Exact rational calculations and precision-refined evaluations illustrate the proved bounds and distinguish a fixed cancellation profile from the growing length of the sum.
ER  -

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