Envelope Bound for the Self-Referential Remainder
Abstract
The additive correction$$A_T = \sum_{t 1\),$$A_T \leq \frac{1}{3}\, T^{1-1/r} \bigl((1-2^{-r})\,\zeta(r)\bigr)^{1/r}.$$The case \(r = 2\) recovers the \(L^2\) estimate \(A_T = O(\sqrt{T})\), while the asymptotically optimal choice \(r_T = 1 + 1/\log T\) gives$$A_T \leq \frac{e}{6}\,\log T + O(\log\log T),$$via the pole of \(\zeta(s)\) at \(s = 1\). This provides a non-circular logarithmic bound on \(A_T\), sufficient for the drop criterion at large \(T\) whenever the co-length surplus is linearly positive. Serie I · v0.4 · DOI all versions: 10.5281/zenodo.19937295
Fields of Study & Subjects
Markov Chains and Monte Carlo Methods
Stochastic processes and statistical mechanics
Distributed systems and fault tolerance
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 2 May 2026
- Language
- EN
- Document type
- Preprint