bibkey: luca2009eulerfibonacci authors: Florian Luca; V. Janitzio Mejía Huguet; Florin Nicolae year: 2009 title: On the Euler Function of Fibonacci Numbers doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL12/Mejia/luca31.pdf claim: Lemma 3 bounds the sum of reciprocal primes with Fibonacci rank m by O(log(m)/m); it does not state a Robin bound for arbitrary two-term sums. strata_touched: [] license: citation-only triage: anchor

Fibonacci rank buckets and Euler-function distributions

The primary article is in Journal of Integer Sequences 12 (2009), Article 09.6.6. Lemma 3, equation (5), page 4, with proof continuing on page 5, states that the sum of 1/p over primes whose first Fibonacci zero index is m is O(log(m)/m). It uses p = ±1 mod m, a cardinality bound from the product of these primes dividing F_m, and a split at m^2. No explicit constant six is stated there. Lemma 4 on page 5 bounds the divisor sum of log(d)/d by O((log log m)^2), citing Luca’s earlier work.

Theorem 1 and the discussion immediately after it on page 2 concern density of vectors of ratios of Euler functions at consecutive Fibonacci numbers. Page 13, Section 6 states the corresponding extension to the divisor-sum function. Neither statement gives an upper bound for the divisor sum of an arbitrary additive combination of Fibonacci terms.

FIB theory §171 compares this established rank-bucket method with the explicit logarithmic Euler estimate in §163 and the common-source carrier bounds in §§167–169. Those combinations are project deductions, not claims attributed to this paper. This note records the verified primary statements, not an exhaustive priority search or a new proof of RH.