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Benoit Cloitre

Publications and source records attributed to Benoit Cloitre.

At least 19 recordsLinked to original sources
arXiv2026

Regular Arithmetic Functions, Volume I. Theory, Applications, Examples

This is the first of two volumes on regular arithmetic functions, and an introduction to their theory. A regular arithmetic function (RAF) is a kernel whose properties come from a single defining equation. Let $G(n,k)$ be a function of two integer variables with $G(n,n)\neq 0$, and for each $\beta$ define a sequence $(a_k)$ by $\sum_{k\le n} a_k G(n,k)=n^{-\beta}$, solved rank by rank with nothing to assume and no convergence to establish. Then $G$ is regular when the partial sums $\sum_{k\le n} a_k$ change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index $\alpha(G)$, a quantity belonging to the kernel itself. The index came out of analogies, experiment and observation, and it is arithmetic by nature. Its most visible application is the Riemann hypothesis, which holds if and only if Ingham's kernel $G(n,k)=(k/n)\lfloor n/k\rfloor$ is a RAF of index $1/2$. It is not the only one. On a problem of a quite different kind the same theory improves the known decay bound for the orthorecursive expansion of unity. Seventeen kernels are worked out in a gallery, so that the notion can be handled by the reader. What is conjectural, conditional or open is marked as such and collected in a register. Volume II is given over to that kernel alone, with its connection to the world of Hasse-Weil zeta functions through a gauged system.

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arXiv2026

Fibonacci, Dirichlet, and Gauss in a single sum

We study the fractional-part sums $\sum_{k=1}^{n}\{F_n/F_k\}$, where $F_n$ is the $n$th Fibonacci number. Their asymptotic behavior depends on the parity of $n$. For odd $n$, the remainder is expressed in terms of the Gauss circle error term. For even $n$, it is expressed in terms of the Dirichlet divisor error term. Thus determining the optimal remainder exponent for the odd Fibonacci sums is equivalent to the Gauss circle problem, while the corresponding question for the even sums is equivalent to the Dirichlet divisor problem. We also prove analogous formulas for a family of second-order recurrences, including the Lucas sequence, for which the roles of the two parities are exchanged.

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arXiv2026

Beatty solutions of almost Golomb functional equations

We study the almost Golomb equation of order $r$, $a\bigl(S(n)\bigr)=n$, where $S(n)=a(n)+a(n-1)+\cdots+a(n-r+1)$, for nondecreasing sequences of positive integers. Its greedy solution is $r$-regular in the sense of Allouche and Shallit. Beyond that one, and for every non-square order $r$, the equation has another solution, an inhomogeneous Beatty sequence $a(n)=\lfloor n/\sqrt{r}+d \rfloor$ where $d$ is a parameter. No other positive slope occurs, and the equation holds for all $n\ge r$ exactly when $d$ lies in an explicit closed interval depending on $r$. That interval is a single point when $r=2$ and has positive length for every non-square $r\ge 3$. Iterating the equation gives, for $k\ge 1$, $a^{\circ k}\bigl(S(n)\bigr)=a^{\circ(k-1)}(n)$, with $a^{\circ 0}$ the identity. Its Beatty solutions of positive slope are again of the form $\lfloor n/\sqrt{r}+d \rfloor$, and the sets of admissible $d$ form an increasing chain. The equation for $k$ holds exactly when $a^{\circ(k-1)}$ agrees at $n$ and at $a(S(n))$. We determine these solutions for $k=2$ when $r=2$ and $r=3$. At the right endpoint of the $r=2$ interval the first equation fails on a thin set of indices, which we identify as the return times of an irrational rotation to an explicit interval. The proofs combine equidistribution with an exact computation in $\mathbb{Z}[\sqrt{r}]$ over a finite range of orders.

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arXiv2026

The Thue-Morse Transform

We define the Thue-Morse transform T on a class of infinite binary words. It sends the alternating word a_0 = 010101... to the Thue-Morse sequence. We then study its orbit a_m = T^m(a_0) as well as the sequences u_m and v_m giving respectively the positions of the ones and the zeros in a_m. We obtain an explicit formula for a_m and deduce Prouhet-Tarry-Escott identities for the partitions induced by u_m and v_m. We also give composition formulas for u_m and v_m, and a full description of the factor complexity of a_m.

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arXiv2026

The Mantovanelli-Hofstadter Sequence

We study the perturbed Hofstadter recurrence introduced by Mantovanelli, $\widetilde{Q}(n)=\widetilde{Q}(n-\widetilde{Q}(n-1))+\widetilde{Q}(n-\widetilde{Q}(n-2))+(-1)^n$, with $\widetilde{Q}(1)=\widetilde{Q}(2)=1$. We prove that this recurrence is well-defined for every positive integer and that $\lim_{n\to\infty}\widetilde{Q}(n)/n=1/2$. We establish the optimal order $\widetilde{Q}(n)=n/2+O(n/\sqrt{\log n})$ and give explicit positive lower and upper bounds for the corresponding normalized limsup. The proof is primarily combinatorial. An odd-even split turns the recurrence into two exact interleavings of binary words, and the resulting Dyck paths define plane forests. Catalan numbers also arise in the enumeration.

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arXiv2026

Almost Golomb Sequences

Golomb's sequence is the unique nondecreasing sequence of positive integers in which each $n$ appears exactly $a(n)$ times. It satisfies the global self-referential rule \[ a\bigl(a(n)+a(n-1)+\cdots+a(1)\bigr)=n, \] grows smoothly like a power of $n$ governed by the golden ratio, and is not $k$-regular for any $k\ge 2$. We introduce almost Golomb sequences, obtained by truncating the cumulative sum to a sliding window of fixed size $r$, \[ a\bigl(a(n)+a(n-1)+\cdots+a(n-r+1)\bigr)=n. \] This finite-memory truncation changes the nature of the sequence completely. The smooth power law gives way to oscillatory linear growth, and the sequence becomes $r$-regular for every $r\ge 2$. For small values of $r$ we establish explicit denesting formulas, prove that $a(n)/n$ does not converge, and uncover combinatorial structure including a cellular automaton and a palindromic substitution. A numerical surprise emerges when one varies $r$. The maximum multiplicity across the family of sequences is governed by Golomb's sequence itself. The sequence that was truncated reappears as the law controlling the family it generated.

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arXiv2026

Meta-automatic Sequences

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's Q-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We exhibit two binary meta-automatic sequences $M_1$ and $M_2$ whose defining recurrences do not satisfy the Allouche-Shallit automaticity criterion directly, and this is formalized in our paper. For each of these integer sequences $M_1$ and $M_2$, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences.

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arXiv2026

The Golden Sieve

We revisit the golden sieve, a self-referential deletion process on increasing sequences of positive integers introduced by the author in 2002. Applied to the natural numbers, the sieve produces the Wythoff pair as a Beatty partition. For arithmetic progressions $a\mathbb{N}+b$, we establish a connection with the $(j,x,y,z)$-hiccup sequences recently studied by Fokkink and Joshi and with Fraenkel's complementary partitions. We further introduce an extraction sieve that also produces hiccup sequences, and whose action on arithmetic progressions is governed by an explicit affine transformation of hiccup parameters.

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arXiv2026

The holonomic triangle: from a symmetry between $e$ and $\pi$ to additive Gamma functions

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational coefficients. We call such functions additive Gamma functions (AGFs), recognizing Euler's $\Gamma(z)$ as the order-1 prototype. Our theory reveals a structural dichotomy: one AGF is expressible as Gamma ratios (regular case), another involves incomplete Gamma (irregular case). AGFs complete a holonomic triangle between P-recursive sequences, additive functional equations, and differential equations, unifying discrete and continuous perspectives under the condition that Gamma factors in asymptotics have integer slopes.

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arXiv2025

Primes in LCM recurrences

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a strengthening of Linnik's theorem on the least prime in an arithmetic progression that lies beyond current reach. We develop a Companion--Sieve framework that reduces the conjecture to an equidistribution problem for primes in the progression $-1\bmod p$, and applying the Bombieri--Vinogradov theorem we prove unconditionally that the conjecture holds for a set of integers of asymptotic density $1$. We also give an effective finite reduction showing that any counterexample beyond a computable threshold involves only large prime factors. A closely related recurrence turns out to encode twin prime pairs through its increment pattern, and we prove a conditional density-$1$ result for it under a prime-index detection hypothesis, using an upper-bound Selberg sieve estimate for twin primes in arithmetic progressions. The analysis also leads to three new conjectures on the distribution of primes in arithmetic progressions.

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arXiv2025

On a BBP-type formula for $\pi^2$ in the golden ratio base

This paper presents a detailed, self-contained proof of a BBP-type formula for $\pi^2$ expressed in the golden ratio base, $\phi$. The formula was discovered empirically by the author in 2004. The proof presented herein is built upon a fundamental geometric identity connecting $\phi$ to the fifth roots of unity, offering an intuitive and direct path to the result. The power of the underlying methodology is then demonstrated by extending it to establish a new, computationally efficient Machin-like formula for $\zeta(3)$, expressed through rapidly converging, hierarchical series involving the golden ratio.

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arXiv2025

An effective analytic recurrence for prime numbers

The Golomb--Keller formula expresses the next prime $p_{n+1}$ as a recurrence relation in terms of the first $n$ primes $p_1, \ldots, p_n$ using the Riemann zeta function and an Euler product, but requires taking a limit as $s \to \infty$, rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter $s \leq p_n$ suffices when combined with the ceiling function, establishing a constructive method valid for all $n \geq 1$. The minimal integer parameter $s_n$ (OEIS A389650) reveals deep connections to prime constellations. We prove $\liminf_{n\to\infty} \sigma_n = 0$ unconditionally, where $\sigma_n = s_n/p_n$. The limit superior $C = \limsup \sigma_n$ satisfies $\log \psi \lesssim C \leq 0.4332$, where $\psi \approx 1.46557$ is the supergolden ratio. The lower bound is conditional on the twin prime conjecture; the upper bound is unconditional. The constant $C$ relates to the densest admissible prime constellation, connecting to the Hardy--Littlewood conjectures. The method extends to Dirichlet L-functions, yielding other effective formulas for calculating $p_{n+1}$ but also for predicting residues of $p_{n+1}$ modulo any integer with reduced precision requirements.

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arXiv2025

A study of a family of self-referential sequences

We introduce and analyze a three-parameter family of self-referential integer sequences $S(x,y,z)$: starting from $a(1)=x$, each term advances by $y$ when the index $k$ has already appeared as a value and by $z$ otherwise. This simple rule generates a surprising zoo of behaviors, many of which are catalogued - albeit in a rather unstructured fashion - in the OEIS. This family has recently and independently been studied by Fokkink and Joshi, who named them "hiccup sequences" and established their general morphic nature. Our work provides a complementary, in-depth analysis of major subfamilies. Whenever $y>z>0$, we prove that the density $a(k)/k$ converges to the positive root of $r^{2}-zr-(y-z)=0$. Two subfamilies, $S(x,Z+1,Z)$ and $S(x,Z,Z+1)$, yield explicit non-homogeneous Beatty sequences, providing explicit formulas for numerous OEIS entries. For $y=0$ and $z \ge 2$, the sequences eventually become periodic and satisfy linear recurrences. Critical cases with a zero discriminant unveil geometric patterns on triangular, square, and hexagonal lattices. Finally, via tree-like representations we uncover a tight link with meta-Fibonacci recurrences. These results position $S(x,y,z)$ as a unifying framework connecting additive combinatorics, number theory, and discrete dynamics.

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arXiv2025

On a Family of Nested Recurrences and Their Arithmetical Solutions

A family of nested recurrence relations $a(n+1) = n - a^{(m)}(n) + a^{(m+1)}(n)$, parameterized by an integer $m \ge 1$ with initial condition $a(1)=1$, is studied. We prove that $a(n)=n-h(n)$ is the unique solution satisfying this condition, where $h(n)$ is an arithmetical sequence in which each non-negative integer $k$ appears $mk+1$ times, with $h(n)$ 1-indexed such that $h(1)=0$. An explicit floor formula for $h(n)$ (and thus for $a(n)$) is derived. The proof of the main theorem involves establishing a key identity for $h(n)$ that arises from the recurrence; this identity is then proved using arithmetical properties of $h(n)$ and the iterated function $a^{(m)}(n)$ at critical boundary points. Combinatorial interpretations for $a(n)$ and its partial sums (for $m=2$), and connections to The On-Line Encyclopedia of Integer Sequences (OEIS), including generalizations of Connell's sequence, are also discussed.

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arXiv2025

On the Orthorecursive Expansion of Unity

The orthorecursive expansion of unity with respect to the system $\{x, x^2, x^3, \ldots\}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds $c_n = O(n^{-3/2})$ and $C_N = \sum_{k=0}^{N} c_k = O(N^{-1/2})$ through intricate $L^2$-norm arguments, but left the optimal decay rates as open problems. We prove $C_N = O_{\varepsilon}(N^{-\alpha_1+\varepsilon})$, where $\alpha_1 \approx 1.3465$ is the smallest real part among the zeros of a transcendental function related to the digamma function. We also improve the coefficient bound to $c_n = O(n^{-2})$. The method rests on a Tauberian transfer theorem that recasts the discrete recurrence as a Volterra integral equation, whose resolvent is smooth and amenable to Mellin analysis and contour shifting.

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arXiv2024

A Tauberian characterization of the Riemann hypothesis through the floor function

We introduce a new Tauberian framework through the theory of "regular arithmetic functions". This allows us to establish a characterization of the Riemann hypothesis by linking the floor function to the distribution of nontrivial zeros of the Riemann zeta function. We thereby obtain a novel Tauberian equivalence of the Riemann hypothesis, extending classical Tauberian theorems beyond their traditional confinement to the prime number theorem. We further uncover connections to combinatorial number theory and set the groundwork for a "combinatorial Tauberian theory", highlighting the broader applicability of regular arithmetic functions.

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arXiv2023

Some Fibonacci-Related Sequences

We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the Walnut theorem-prover. We also prove two new results about synchronized sequences.

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