Some properties of the class of arithmetic functions
R. P. Pakshirajan (1963)
Annales Polonici Mathematici
Similarity:
R. P. Pakshirajan (1963)
Annales Polonici Mathematici
Similarity:
Antonio M. Oller-Marcén (2017)
Mathematica Bohemica
Similarity:
A homothetic arithmetic function of ratio is a function such that for every . 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 in terms of the period and the ratio of .
José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2023)
Commentationes Mathematicae Universitatis Carolinae
Similarity:
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 with the property that every prime number that divides also divides , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are...
Atsushi Moriwaki (2014)
Annales de la faculté des sciences de Toulouse Mathématiques
Similarity:
In this paper, we give a numerical characterization of nef arithmetic -Cartier divisors of -type on an arithmetic surface. Namely an arithmetic -Cartier divisor of -type is nef if and only if is pseudo-effective and .
Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)
Czechoslovak Mathematical Journal
Similarity:
Let and . Denote by the set of all integers whose canonical prime representation has all exponents being a multiple of or belonging to the arithmetic progression , . All integers in 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. ...
Przemysław Mazur (2015)
Acta Arithmetica
Similarity:
We prove that every set A ⊂ ℤ satisfying 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 .
Enrique González-Jiménez (2015)
Acta Arithmetica
Similarity:
Let and a,q ∈ ℚ. Denote by the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve . We study the set and we parametrize it by the rational points of an algebraic curve.
Jan Krajíček (2001)
Fundamenta Mathematicae
Similarity:
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...
Roman Ger, Tomasz Kochanek (2009)
Colloquium Mathematicae
Similarity:
We show that any quasi-arithmetic mean and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations and are the constant ones.
Janusz Matkowski (2013)
Colloquium Mathematicae
Similarity:
A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions , k ≥ 2, denoted by , is considered. Some properties of , 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...
J. S. Ratti, Y. -F. Lin (1990)
Colloquium Mathematicae
Similarity:
Melvyn B. Nathanson, Kevin O'Bryant (2015)
Acta Arithmetica
Similarity:
A geometric progression of length k and integer ratio is a set of numbers of the form for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence of positive real numbers with a₁ = 1 such that the set contains no geometric progression of length k and integer ratio. Moreover, is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that...
Zofia Adamowicz, Konrad Zdanowski (2011)
Fundamenta Mathematicae
Similarity:
We prove that for i ≥ 1, the arithmetic does not prove a variant of its own Herbrand consistency restricted to the terms of depth in , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in .
A. Alaca, Ş. Alaca, E. McAfee, K. S. Williams
Similarity:
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 , where n ∈ ℕ and F: ℤ → ℂ is an even function, is equivalent to the Lambert series for (θ ∈ ℝ, |q| < 1) given by Ramanujan.
Bakir Farhi (2013)
Colloquium Mathematicae
Similarity:
We present new structures and results on the set of mean functions on a given symmetric domain in ℝ². First, we construct on a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on a structure of metric space under which is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of . Finally, we give...
Paweł Pasteczka (2016)
Colloquium Mathematicae
Similarity:
We work with a fixed N-tuple of quasi-arithmetic means generated by an N-tuple of continuous monotone functions (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping tend pointwise to a mapping having values on the diagonal of . Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means taken...
Neha Prabhu (2017)
Czechoslovak Mathematical Journal
Similarity:
A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly prime factors for . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree with prime factors such that a fixed quadratic equation has exactly solutions modulo . ...
Karin Halupczok (2013)
Journal de Théorie des Nombres de Bordeaux
Similarity:
Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd , Goldbach’s ternary problem is solvable with primes in short intervals with , , and such that has at most prime factors.
Oleg Petrushov (2015)
Acta Arithmetica
Similarity:
We consider the behavior of the power series as z tends to along a radius of the unit circle. If β is irrational with irrationality exponent 2 then . Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that .
Zofia Adamowicz (2002)
Fundamenta Mathematicae
Similarity:
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 , where 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.
Taras Banakh, Vesko Valov
Similarity:
General position properties play a crucial role in geometric and infinite-dimensional topologies. Often such properties provide convenient tools for establishing various universality results. One of well-known general position properties is DDⁿ, the property of disjoint n-cells. Each Polish -space X possessing DDⁿ contains a topological copy of each n-dimensional compact metric space. This fact implies, in particular, the classical Lefschetz-Menger-Nöbeling-Pontryagin-Tolstova embedding...