arXiv subjects

Marco Mantovanelli

Publications and source records attributed to Marco Mantovanelli.

9 recordsLinked to original sources

arXiv2026

The Right Edge of the Zero Set of the Fibonacci Zeta Function

Let $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$, and define the Fibonacci zeta function by $$ Z_F(s)=\sum_{n\ge1}F_n^{-s},\qquad \operatorname{Re}s>0. $$ We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If $\sigma_F$ is the unique solution of $$ Z_F(\sigma_F)=4+2\,144^{-\sigma_F}, $$ then $$ \sigma_F=0.743163398726901648\ldots, $$ $Z_F(s)\neq0$ for $\operatorname{Re}s\ge\sigma_F$, while $$ \overline{\{\operatorname{Re}\rho:Z_F(\rho)=0,\ \operatorname{Re}\rho>0\}}=[0,\sigma_F]. $$ The edge is sharp in an almost-periodic sense: zeros occur with relatively dense ordinates near every admissible vertical line. We prove growing-dimensional phase locking near the edge and a Diophantine zero-free cusp, and describe the associated Jessen function and smooth mean vertical zero density. For every partial sum with $N\ge12$ we determine the corresponding exact closure edge $\sigma_N$, prove $\sigma_N\nearrow\sigma_F$, and obtain an exponential asymptotic for $\sigma_F-\sigma_N$. We also derive a finite-core theorem for positive integral Lucas zeta functions, with the Pell zeta function as an explicit example. Finally, using the known meromorphic continuation, we construct a natural $q$-Pochhammer completion that is entire of exact order $2$ and type $\log\varphi/4$.

math.NT↗

arXiv2026

A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers

We study the parity-perturbed Hofstadter $Q$-recursion $$ \widetilde Q(1)=\widetilde Q(2)=1,\qquad \widetilde Q(n)=\widetilde Q(n-\widetilde Q(n-1)) +\widetilde Q(n-\widetilde Q(n-2))+(-1)^n, $$ and the associated two-variable Dirichlet series $$ Z_{\widetilde Q}(s,t)=\sum_{n\ge1}n^{-s}\widetilde Q(n)^{-t}. $$ The estimate $\widetilde Q(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\operatorname{Re}(s+t)>1$. With $w=s+t$, we separate the universal term $2^t\zeta(w)$ and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns $t=-1$. For $E(n)=2\widetilde Q(n)-n$ and $A(X)=\sum_{n\le X}E(n)$, the binary-arch clock yields $$ A(X)=X\log_2X+X\Omega\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right), $$ where $\Omega$ is an explicit continuous periodic function. This continues the normalized correction to $\operatorname{Re}w>0$ and yields a boundary resonance lattice: a double resonance at $w=0$ and simple resonances at $2\pi i m/\log2$. After subtracting the full-slice order-$X$ skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$ A_r=\frac{512}{9\sqrt{2\pi}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right). $$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.

math.NT↗

arXiv2026

Prime-Exponent Transition Geometry and Divisor Barriers Between Consecutive Highly Composite Numbers

Let $d(n)$ be the divisor function and let $H<H'$ be consecutive highly composite numbers. We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H'$. The normalised capacity of such a geodesic is its smallest divisor count divided by $d(H)$, giving a finite fixed-endpoint maximin problem. The exact record enumeration first finds a failure of the static surrogate $d(\gcd(H,H'))\ge d(H)/2$ at $48,886,437,600<64,250,746,560$, where the ratio is $4/9$. For every state $z$ in the exponent box, however, we prove $$d(z)d(HH'/z)\ge d(H)d(H')$$ and deduce the record-box gap: no box state lies numerically strictly between the two records. We also give an exact dynamic-programming recursion, solve the strata $L_-\le 1$, and reduce the complete $L_-=2$ problem to an explicit divisor-selection functional. A computer-assisted enumeration through $10^{70}$ produces $889$ records and $888$ transitions. Although the static half-gcd bound fails $119$ times, every computed geodesic capacity is at least $1/2$; equality occurs in exactly the $124$ transitions that lose an exponent-one support prime. Independently checked certificates cover all $301$ transitions with $L_-=2$. The corresponding universal half-capacity bound and equality classification remain open beyond the proved strata and the verified range. COMMENTS

math.NT↗

arXiv2026

Certified Minimal-Prime Branch Closures for Odd Perfect Numbers

For an odd perfect number $N$, write $q=\min\{p:p\mid N\}$ for its smallest prime divisor. This paper proves a certified branch-closure theorem for the five minimal-prime branches $q=5,7,11,13,17$. The proof combines the exact $q$-adic valuation balance for $\sigma(N)=2N$ with lower-prime avoidance: primes below $q$ cannot occur in the support and therefore cannot divide any divisor-sum factor. These constraints reduce each branch to a finite first-input coverage split and then to terminal forced-or-pure cofactor records. The terminal records are checked by the frozen certificate release C-small-2026-07, consisting of JSONL certificate bundles, Python verifier scripts, expected terminal outputs, and SHA256 hashes. The $q=5$ branch is presented as the detailed audit model, while the branches $q=7,11,13,17$ are closed by the same forced-or-pure mechanism. The result is scoped: it does not prove nonexistence of odd perfect numbers, and the branches $q=3$ and $q\ge 19$ remain outside the paper.

math.GM↗

arXiv2026

Averaged Extensions of Golomb's Triangular Recursion: Critical Invariance and Supercritical Constraints

For an integer $m\ge 1$ and a parameter $\alpha>0$, consider the nested recursion $$ Q_{\alpha,m}(n)=1+\left\lfloor \frac{\alpha}{m}\sum_{j=1}^m Q_{\alpha,m}\!\left(n-Q_{\alpha,m}(n-j)\right)\right\rfloor, \qquad n>m, $$ with $Q_{\alpha,m}(1)=\cdots=Q_{\alpha,m}(m)=1$. For $m=1$ and $\alpha=1$, this is Golomb's non-homogeneous triangular recursion. We prove that its canonical triangular solution is preserved, up to an initial index shift, by every finite arithmetic averaging length. More generally, the same exact solution is generated by any aggregator satisfying a local floor-lock condition. This class includes all power means of finite order, including the harmonic and geometric means, as well as the minimum and positively weighted quasi-arithmetic means. For $m\ge 2$, the maximum lies outside this class but has a different explicit block law. Consequently, every value $k\ge 2$ occurs exactly $k$ times in the floor-admissible class, and $$ Q_{1,m}(n)=\left\lfloor\frac{1+\sqrt{1+8(n-m)}}{2}\right\rfloor \sim\sqrt{2n}. $$ For $0<\alpha<1$, the sequence is identically one. Near criticality, with $\alpha=1+\delta$ and $0<\delta<(2m-1)^{-1}$, we determine the exact first departure time from the critical orbit, of order $\delta^{-2}$. We also prove a finite-step breakdown criterion for large $\alpha$ and a conditional slope theorem: any globally defined solution with a limiting density in $(0,1)$ must have slope $1-\alpha^{-1}$. Exact-arithmetic computations support, but do not prove, a supercritical linear-growth regime.

math.CO↗

arXiv2026

The Subtractive Divisor Orbit: Unconditional Bounds, Parity Constraints, and a Conditional Framework

Let $\tau(n)$ denote the number of positive divisors of $n$. Starting from $n_0=x$, consider the orbit $n_{j+1}=n_j-\tau(n_j)$, and let $a(x)$ be its hitting time of zero. Although the average order of $\tau$ suggests $a(x)\asymp x/\log x$, the orbit samples the divisor function endogenously, and no unconditional estimate of this order is known to us. We prove the exact identity $\sum_{j<a(x)}\tau(n_j)=x$ and the unconditional bounds $$ \frac{x}{(\log(2x))^3}\ll a(x)\le \frac{3x}{8}+O\!\left(\frac{\sqrt{x}}{\log x}\right). $$ We also show that the orbit changes parity exactly at square states. On dyadic orbit segments, we establish a local-to-global criterion, a large-value truncation, and a quantitative implication from small relative variance to a step-mass-saturating dynamic near-ladder. Finally, under two explicit hypotheses -- a regularity-or-ladder dichotomy and an anti-ladder estimate -- we obtain $a(x)\asymp x/\log x$.

math.NT↗

arXiv2026

Certified Return-Word Induction for a Perturbed Hofstadter Recursion

We study the parity-perturbed Hofstadter recursion $$ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n. $$ We prove that it is well-defined for every $n\ge 1$ by a computer-assisted return-word induction. The recurrence is first reduced to a binary sequence $s_n$ together with two backward cursor heads. Its local semantics is encoded by 13 return-word types, 92 synchronized cursor states, and 122 exact two-source transition rules. Exhaustive finite checks verify the local recurrence identity, cursor synchronization, rule selection, and exclusion of the unique configuration that could produce a nonbinary value. The global argument is not inferred from a long finite trace. Instead, all unbounded word families are handled by explicit induction: four stationary chunk families and four linear bridge-tail families reduce to ten parameterized zero-loop schemas. Ordered rank-word factorizations assemble complete epochs and bridges at every level. A marked factor induction then propagates absolute source-block addresses through the resulting four-factor cycle, while a finite potential certificate yields the uniform lag bounds $$ j-p_A\ge 38,\qquad j-p_B\ge 38, $$ so every source read lies strictly in the previously generated prefix. Finally, defining $Q(n)=n+1-T_{n+1}$ from the constructed binary system gives a two-periodic recurrence residual that vanishes in the two base cases. Hence the original recursion holds globally, all recursive arguments are positive and strictly smaller than the current index, and the sequence is uniquely determined.

math.CO↗

arXiv2026

Quadratic Reduction and Classical Multiple-Index Fibonacci-Lucas Identities

We place several classical identities in a common algebraic framework. Let $x$ be an element of a unital algebra over a commutative ring $R$ satisfying $$ x^2-tx+d1=0. $$ Then every positive power of $x$ admits the universal reduction $$ x^m=P_m(t,d)x-dP_{m-1}(t,d)1, $$ where the coefficient polynomials $P_m$ form the generic Lucas sequence and are equivalently Dickson polynomials of the second kind. By the Cayley--Hamilton theorem, this gives the corresponding standard formula for powers of $2\times2$ matrices, whose scalar coefficients depend only on the trace and determinant. Applying suitable $R$-linear functionals to powers of the Fibonacci matrix yields uniform derivations of multiple-index identities for Fibonacci, Lucas, and generalized Fibonacci sequences. In particular, we recover the expansion for $F_{nm}$ derived by Mc Laughlin, who attributes it to Johnson, and recently reproved by Vorobtsov. The purpose of this note is expository: to make explicit the common mechanism connecting quadratic reduction, Lucas--Dickson polynomials, and matrix methods.

math.CO↗

arXiv2026

A Dyadic Frequency Law for a Perturbed Hofstadter $Q$-Recursion

We study the perturbed Hofstadter $Q$-recursion $$ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n \quad (n\ge 3). $$ Clo\^itre proved that the recursion is globally well-defined and encoded its odd- and even-indexed subsequences by exact binary arches and canonical plane forests. Since every value is odd, let $$ F(s)=\#\{n\ge 1:Q(n)=2s-1\}. $$ We prove that, for every $k\ge 0$, $$ \{F(s):2^k\le s<2^{k+1}\} = \{3+\nu_2(j):1\le j\le 2^k\} $$ as multisets. Thus the theorem determines the multiplicities of the frequencies in each dyadic block, but not their order. The proof converts frequencies into plateau local times, folds paired gap degrees under reversal-complementation, and identifies the resulting multiset with the degree multiset of a canonical tree. An ordered central-pair lemma is the boundary step that makes the dyadic cut exact.

math.CO↗