A note on $(a,b)$-Fibonacci sequences and specially multiplicative arithmetic functions

Emil Daniel Schwab; Gabriela Schwab

Displaying similar documents to “A note on $(a, b)$ -Fibonacci sequences and specially multiplicative arithmetic functions”

Herbrand consistency and bounded arithmetic

Zofia Adamowicz (2002)

Fundamenta Mathematicae

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We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove $T ₘ ⊬ H C o n s^{I ₘ} (T ₘ)$ , where $H C o n s^{I ₘ}$ is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.

On integers of the form $p + 2^{k}$

Laurent Habsieger, Xavier-François Roblot (2006)

Acta Arithmetica

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Aposyndesis in $ℕ$

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2023)

Commentationes Mathematicae Universitatis Carolinae

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We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions $P (a, b)$ with the property that every prime number that divides $a$ also divides $b$ , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are...

Lambert series and Liouville's identities

A. Alaca, Ş. Alaca, E. McAfee, K. S. Williams

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The relationship between Liouville’s arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville’s arithmetic formula for the sum $\sum_{\binom{(a, b, x, y) \in ℕ ⁴}{a x + b y = n}} (F (a - b) - F (a + b))$ , where n ∈ ℕ and F: ℤ → ℂ is an even function, is equivalent to the Lambert series for $(\sum_{n = 1}^{\infty} (q ⁿ / (1 - q ⁿ)) s i n n θ) ²$ (θ ∈ ℝ, |q| < 1) given by Ramanujan.

Some properties of the class of arithmetic functions $T_{r} (N)$

R. P. Pakshirajan (1963)

Annales Polonici Mathematici

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Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov (2014)

Acta Arithmetica

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Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound $O (n^{3 / 2})$ .

An inconsistency equation involving means

Roman Ger, Tomasz Kochanek (2009)

Colloquium Mathematicae

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We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f (M (x, y)) = A_{φ} (f (x), f (y))$ and $f (A_{φ} (x, y)) = M (f (x), f (y))$ are the constant ones.

Numerical characterization of nef arithmetic divisors on arithmetic surfaces

Atsushi Moriwaki (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper, we give a numerical characterization of nef arithmetic $ℝ$ -Cartier divisors of $C^{0}$ -type on an arithmetic surface. Namely an arithmetic $ℝ$ -Cartier divisor $\bar{D}$ of $C^{0}$ -type is nef if and only if $\bar{D}$ is pseudo-effective and $\hat{\deg} ({\bar{D}}^{2}) = \hat{vol} (\bar{D})$ .

A note on Δ₁ induction and Σ₁ collection

Neil Thapen (2005)

Fundamenta Mathematicae

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Slaman recently proved that Σₙ collection is provable from Δₙ induction plus exponentiation, partially answering a question of Paris. We give a new version of this proof for the case n = 1, which only requires the following very weak form of exponentiation: " $x^{y}$ exists for some y sufficiently large that x is smaller than some primitive recursive function of y".

On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

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A homothetic arithmetic function of ratio $K$ is a function $f : ℕ \to R$ such that $f (K n) = f (n)$ for every $n \in ℕ$ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of $f (ℕ)$ in terms of the period and the ratio of $f$ .

On the weak pigeonhole principle

Jan Krajíček (2001)

Fundamenta Mathematicae

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We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs of the Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions...

A functional equation involving f and $f^{- 1}$ .

J. S. Ratti, Y. -F. Lin (1990)

Colloquium Mathematicae

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On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

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Let $m \in ℤ_{> 0}$ and a,q ∈ ℚ. Denote by $_{m} (a, q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_{d} : x ² + y ² = 1 + d x ² y ²$ . We study the set $_{m} (a, q)$ and we parametrize it by the rational points of an algebraic curve.

Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Janusz Matkowski (2013)

Colloquium Mathematicae

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A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f ₁, . . ., f_{k} : I \to ℝ$ , k ≥ 2, denoted by $A^{[f ₁, . . ., f_{k}]}$ , is considered. Some properties of $A^{[f ₁, . . ., f_{k}]}$ , including “associativity” assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For...

A structure theorem for sets of small popular doubling

Przemysław Mazur (2015)

Acta Arithmetica

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We prove that every set A ⊂ ℤ satisfying $\sum_{x} m i n (1_{A} * 1_{A} (x), t) \leq (2 + δ) t | A |$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ (| ℕ ∖ (A + A) | \geq k) = Θ (2^{- k / 2})$ .

О моделях арифметики в $A$ -изолиях.

В.Л. Михеев (1991)

Algebra i Logika

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Multiplicative functions dictated by Artin symbols

Robert J. Lemke Oliver (2013)

Acta Arithmetica

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Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/ℚ, we construct a natural class $_{K}$ of completely multiplicative functions whose values are dictated by Artin symbols, and...

General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, Štefan Porubský (2009)

Studia Mathematica

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In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d} * g^{* d} + a_{d - 1} * g^{* (d - 1)} + \dots + a ₁ * g + a ₀ = 0$ , where $a ₀, . . ., a_{d} : ℕ \to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form $\sum_{x \in X} f (x) e^{- s x}$ ( $s \in ℂ^{k}$ ), where $X \subseteq {[0, \infty)}^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion...

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

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Let $a$ and $b \in ℕ$ . Denote by $R_{a, b}$ the set of all integers $n > 1$ whose canonical prime representation $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{r}^{α_{r}}$ has all exponents $α_{i}$ $(1 \leq i \leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $a t + b$ , $t \in ℕ_{0} : = ℕ \cup {0}$ . All integers in $R_{a, b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given. ...

Displaying similar documents to “A note on ( a , b ) -Fibonacci sequences and specially multiplicative arithmetic functions”

Displaying similar documents to “A note on $(a, b)$ -Fibonacci sequences and specially multiplicative arithmetic functions”