bibkey: drmotamullnerspiegelhofer2025primes authors: Michael Drmota; Clemens Müllner; Lukas Spiegelhofer year: 2025 title: Primes as sums of Fibonacci numbers doi: 10.1090/memo/1537 url: https://arxiv.org/abs/2109.04068v2 claim: “Theorem 2.4 gives level-one distribution for the Zeckendorf digit-sum phase, with a fixed-epsilon modulus range. The full initial-prime support modulus of an actual superabundant candidate eventually lies outside every such fixed range at its own size.” strata_touched: [] license: citation-only triage: anchor

Zeckendorf digital distribution and the full support modulus

Published as Memoirs of the American Mathematical Society 305, memoir 1537 (2025), DOI:10.1090/memo/1537. The theorem inspected here is from the authors’ arXiv:2109.04068v2, updated 2022-08-16, §2.5.1, Theorem 2.4, equation (2.5), printed p.12. The publisher PDF was not available at the inspected endpoint; theorem numbering is therefore assigned to that specified author version. The statement and the parameter comparison below were checked, not the complete analytic proof. The application has no Lean verification and is not claimed as new mathematics or a Robin proof.

What level one means in the quoted theorem

Let $s_{\phi }(n)$ be the number of occupied positions in the canonical Zeckendorf representation, $e(t)=\exp (2\pi it)$ and $\Vert \theta \Vert ={\min }_{k\in \mathbb{Z}}|\theta -k|$. For each fixed $\epsilon >0$, there exist $c_{\epsilon },C_{\epsilon }>0$, depending only on $\epsilon$, such that for every real $X\ge 1$ and $\theta \in \mathbb{R}$,

$$\sum _{1\le d\le X^{1-\epsilon }}\underset{\begin{array}{c}0\le u\le v\\ v-u\le X\end{array}}{\max }\underset{0\le b<d}{\max }\left|\sum _{\begin{array}{c}u\le n<v\\ n\equiv b(\mathrm{mod}d)\end{array}}e\left(\theta s_{\phi }(n)\right)\right|\le C_{\epsilon }({\log }^{+}X{)}^{11/4}{X}^{1-c_{\epsilon }\Vert \theta {\Vert }^2}.\text{\hspace{1em}(1)}$$

Here ${\log }^{+}X=\max (1,\log X)$. The interval location is uniform, and a fixed admissible modulus can be extracted from the nonnegative outer sum. The theorem still has a fixed positive $\epsilon$: “level one” does not assert a uniform bound through $d=X$, or permit $\epsilon$ to tend to zero without control of its constants. At integral $\theta$, the displayed bound has no power saving; that degenerate phase is included in the statement.

The actual extremal support has a different scale

Reuse Alaoglu–Erdős, On highly composite and similar numbers, §2, Theorems 1 and 7. The inspected original text places Theorem 7 on printed p.454. For superabundant $n$, including colossally abundant integers, these classical inputs give initial-prime support and

$$P=P^{+}(n)\sim \log n,R=P\#=\prod _{p\le P}p\mid n.$$

The prime number theorem then gives

$$\log R=\vartheta (P)\sim \log n,R=n^{1-o(1)}.\text{\hspace{1em}(2)}$$

Thus at $X\asymp n$, the single modulus imposing all divisibilities $p\mid n$ for $p\le P$ eventually satisfies $R>X^{1-\epsilon }$ for every fixed $\epsilon >0$. The range in (1) consequently cannot encode that complete support at the candidate’s own scale. The higher prime-power valuations are additional restrictions, beyond this already too-large support modulus.

This diagnoses the full-support substitution, rather than all possible uses of (1). Smaller subsets of support primes may fall within its range. Taking a larger $X$ can also make $R$ admissible, but (1) then carries the larger absolute error and supplies no estimate below one that isolates the original integer. Uniformity in interval location does not eliminate the dependence on interval-length bound $X$.

The observable must also be transported

The phase in (1) counts all Zeckendorf digits. It is not the complete golden norm, discriminant square class, canonical displacement, or exponent-sensitive Euler deficit of that same $n$. A transfer requires an estimate for the required observable and its weights, not just the availability of a canonical digit encoding. The weighted Beatty note retains the actual unit bit and $Z(n)$, but likewise supplies only an average.

The earlier result Möbius orthogonality for the Zeckendorf sum-of-digits function, Theorem 1, is already cited in the FIB theory, §148.2; it is not supplied again as a new input here. The effective pointwise fixed-digit and fixed-support results of Bugeaud are likewise already recorded in the complementary-divisor note. Neither the moving full-support modulus nor the same-candidate weighted estimate follows from these cited results.