Group automorphisms with prescribed growth of periodic points,and small primes in arithmetic progressions in intervals

We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik?s Theorem, concerning small prime numbers in arithmetic progressions which lie in intervals.     

On the density of integers of the form 2 k + p in arithmetic progressions

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.     

SOME ALMOST ALL RESULTS CONCERNINGTHE PJATECKII-SAPIRO PRIME NUMBERS

51.IntroductionSupposethatr>Oisafixednumberand[oldenotes,asusuaI,theintegralpartofo.LetForo1wasfirststudiedin1953byPjateckii-Sapiro[']whoprovedthat(1.1)holdsfor1相似文献   

Average distribution of k-free numbers in arithmetic progressions

We obtain a new bound on the average value of the error term in the asymptotic formula for the number of k-free numbers in arithmetic progressions. In particular, we improve the results of J. Gibson (2014) and C. Hooley (1975).     

On arithmetic progressions on Pellian equations

We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x ² ? dy ² = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves.     

Arithmetic on Groups of Positive Rationals

In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory of genera in algebraic number fields. We establish generalizations of both Dirichlet's theorem on primes in arithmetic progressions and the theory of genera for Abelian extensions of the rationals.     

On the large sieve in algebraic number fields

In the present note Bombieri's central theorem concerning the average distribution of the prime numbers in arithmetic progressions is generalized to arbitrary algebraic number fields.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 673–680, December, 1967.Finally, I express my profound gratitude to B. V. Levin for setting the problem and the help he rendered and to A. I. Vinogradov for valuable suggestions.     

Character sums over squarefree and squarefull numbers

We give upper bounds for character sums over squarefree and squarefull numbers sharper than the prior known in the literature. As an application, we study the distribution of squarefull numbers in arithmetic progressions and sharpen (without restriction on the modulus) the recent results obtained by Liu and Zhang.     

The closures of arithmetic progressions in the common division topology on the set of positive integers

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.     

Twins of <Emphasis Type="Italic">k</Emphasis>-free numbers in arithmetic progressions

We give a new upper bound of Barban–Davenport–Halberstam type for twins of k-free numbers in arithmetic progressions.     

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.     

Some new results on k-free numbers

In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban-Davenport-Halberstam type for the k-free numbers in arithmetic progressions.     

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.     

On the length of arithmetic progressions in linear combinations of S-units

Recent finiteness results concerning the lengths of arithmetic progressions in linear combinations of elements from finitely generated multiplicative groups have found applications to a variety of problems in number theory. In the present paper, we significantly refine the existing arguments and give an explicit upper bound on the length of such progressions.     

On Irreducibles in the Polynomial Semigroup

We consider the asymptotic behavior of the number of irreducible polynomials over a finite field in arithmetic progressions.     

An upper bound for van der Waerden-like numbers usingk colors

Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions,A, by a classA′, withA ? A′; thus, the associated van der Waerden-like number will be smaller forsi’. We consider increasing sequences of positive integers x₁,…, x_n which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = x_i+1, i = 1,…,n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n,k). A table of several new computer-generated values of these functions is provided.     

On the distribution of square-full numbers in arithmetic progressions

A positive integer n is called a square-full number if p ² divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions.     

Greedy algorithm, arithmetic progressions, subset sums and divisibility

We consider a number of density problems for integer sequences with certain divisibility properties and sequences free of arithmetic progressions. Sequences of the latter type that are generated by a computer using modifications of the greedy algorithm are also provided.     

Connections between connected topological spaces on the set of positive integers

In this paper we introduce a connected topology T on the set ? of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ? which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (?, T) and (?, T′).     

Application of theorems on primes to diophantine problems of a special type

In this paper we consider the problem of whether the equation $$\begin{array}{*{20}c} {n\frac{{v_1 \varphi _1 - v_2 \varphi _2 }}{{v_1 - v_2 }}} & {v_1 \ne v_2 } \\ \end{array} $$ can be solved and of a lower bound for the number of solutions,subject to certain constraints on the density of the numbers ν and the distribution of the numbers ? in arithmetic progressions.