bibkey: guloglunevans2008beatty authors: Ahmet M. Güloğlu; C. Wesley Nevans year: 2008 title: Sums of multiplicative functions over a Beatty sequence doi: 10.1017/S0004972708000853 url: https://arxiv.org/abs/0801.2796v1 claim: “Theorem 1 controls multiplicative-function sums on a finite-type Beatty sequence. Applied to sigma(n)/n and the existing canonical unit-bit bridge, it gives a same-integer weighted branch average, without a bound for an individual extremal candidate.” strata_touched: [] license: citation-only triage: anchor

Multiplicative Beatty sums and the canonical FIB unit bit

Bull. Austral. Math. Soc. 78 (2008), 327–334. The inspected primary text is the published article. Its title uses “of”; arXiv:0801.2796v1 uses “with”. The function class and Theorem 1 agree in these versions. The source statement, parameter conditions and application below were checked; the complete analytic proof was not independently verified. The applications are paper derivations without new Lean declarations, Lean verification or an originality claim.

The source theorem and its fixed function class

Printed p.327 defines, for a fixed $A\ge 1$, the class ${\mathcal{F}}_A$ of multiplicative functions satisfying

$$|f(p)|\le A(p\text{ prime}),\sum _{n\le N}|f(n){|}^2\le A^{2}N(N\in {\mathbb{N}}_{>0}).$$

For $\alpha >1$ and real $\beta$, put $B_{\alpha ,\beta }=\{\lfloor \alpha k+\beta \rfloor :k\in \mathbb{Z}\}$. Theorem 1, printed p.328, assumes that $\alpha$ is irrational of finite Diophantine type and gives

$$\sum _{\begin{array}{c}n\le N\\ n\in B_{\alpha ,\beta }\end{array}}f(n)={\alpha }^{-1}\sum _{n\le N}f(n)+O_{\alpha ,A}\left(\frac{N\log \log N}{\log N}\right).\text{\hspace{1em}(1)}$$

All sums over $n$ use positive integers. The implied constant depends only on $\alpha ,A$, so the statement is uniform in $\beta$ and in $f\in {\mathcal{F}}_A$. It does not remove the dependence on $A$ for a family whose second moments grow with $N$.

Reusing the existing canonical observation

Let $\phi =(1+\sqrt{5})/2$, $\psi =1-\phi$, and $a=2-\phi$. For the actual canonical Zeckendorf representation of $n>0$, let $h(n)\in \{0,1\}$ indicate whether the unit term is present. It is not the occupancy of the numerical term 2. The existing Zeckendorf–Beatty bridge uses Fibonacci index 2, namely $F_2=1$, and the existing mechanical-word window identifies its complement. Their application gives

$$h(n)=1⟺a<\{n\phi \}<2a⟺n=\lfloor k{\phi }^2-1\rfloor \text{for some }k\ge 1.\text{\hspace{1em}(2)}$$

The positive values of $B_{{\phi }^2,-1}$ come exactly from $k\ge 1$. The Beatty correspondence is reused, rather than a new numeration theorem; the Dekking entry already points to the repository’s canonical bridges.

Write $n=h+2A+3B$ using the complete composition $(A,B)$ after removing the unit. Put $c=4A+7B$, $m=\lfloor n\phi \rfloor$, $r=\{n\phi \}$ and $E=c-\sqrt{5}(n-h)$. The existing displacement reading, applied at $n-h$, gives $c=2(\lfloor (n-h+1)\phi \rfloor -1)-(n-h)$. Its integer lifts and residuals are therefore

No positive integer hits an endpoint. The parity condition is retained: $c\equiv n-h(\mathrm{mod}2)$, and the original integer composition is recovered by $B=c-2(n-h)$ and $A=(7(n-h)-3c)/2$. For $n=1$, these formulas give $h=1,A=B=E=0$; any argument requiring nonzero composition must exclude that case separately.

A weighted average for the same arithmetic integer

Take $Z(n)=\sigma (n)/n$. It is multiplicative and $Z(p)=1+1/p\le 3/2$. The source already supplies

$$\sum _{n\le N}\frac{\sigma (n{)}^2}{n^2}=\frac{5}{2}\zeta (3)N+O((\log N{)}^3)$$

in printed p.333, proof of Corollary 4. This implies the required second-moment bound for some fixed $A$: enlarge it to cover the finite initial range. No new second-moment theory is needed. Since the quadratic irrational ${\phi }^2$ has finite type, (1) and (2) yield

$$\sum _{\begin{array}{c}n\le N\\ h(n)=1\end{array}}Z(n)={\phi }^{-2}\sum _{n\le N}Z(n)+O\left(\frac{N\log \log N}{\log N}\right).\text{\hspace{1em}(3)}$$

This keeps the canonical unit bit and the multiplicative response of the same integer. It is not an average over independently selected Fib addresses and factorizations. Nevertheless its allowed error exceeds even the elementary single-term upper bound $Z(n)\le 1+\log n$. It does not bound the maximum within either unit branch. Replacing $Z$ by $Z{1}_{\mathrm{CA}}$ or $Z{1}_{\mathrm{SA}}$ does not preserve the theorem’s multiplicativity hypothesis. Growing moments such as $Z^s$ require a fresh bound for their function-class constant and its effect on the error.

The small-discriminant condition needs shrinking rotation windows

Using the same complete composition, put $Q=A^2+AB-B^2$ and $D=5h^2-4Q$. The existing affine certificate in the FIB theory, §202 gives

$$D=5h^2+E(2\sqrt{5}(n-h)+E).\text{\hspace{1em}(4)}$$

For $n\ge 2$ and $L\ge 0$, $c\ge 2(n-h)$ implies the necessary bound

$$|D|\le L⟹|E|\le \frac{L+5}{(2+\sqrt{5})(n-1)}.\text{\hspace{1em}(5)}$$

Thus at $n\asymp X$ the raw-discriminant condition $|D|\le (\log n{)}^K$ needed by the nonresidue cutoff lies in rotation windows of width $O((\log X{)}^K/X)$. The fixed unit window in (3) is a different resolution. Neither (3) nor (5) forces an actual CA/SA candidate into such a shrinking window, proves nonsquareness, or bounds the squarefree kernel of $D$. Those joint arithmetic conditions remain missing.