On the mean value of the index of composition of an integral ideal

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In this paper, we study the mean value of the index of composition of an integral ideal, and get an asymptotic formula for it.

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On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

On the Mean Value of the Index of Composition of an Integer

For each integer n 2, let be the index of composition of n, where . For convenience, we write (1)=(1)=1. We obtain sharp estimates for and , as well as for and . Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Values of the Euler function free of kth powers

We establish an asymptotic formula for the number of positive integers n?x for which φ(n) is free of kth powers.     

On digit sums of multiples of an integer

Let g>1 be an integer and sg(m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sg(n) and sg(kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sg(n)=sg(kn). In the second part of the paper, we show that for any K>0 the set of the integers n satisfying sg(n)?Ksg(kn) for all k∈N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.     

CONTRIBUTIONS TO ABSTRACT ANALYTIC NUMBER THEORY

Abstract

This paper reports on some recent contributions to the theory of multiplicative arithmetic semigroups, which have been initiated by John Knopfmacher's work on abstract analytic number theory. They concern weighted inversion theorems of the Wiener type, mean-value theorems for multiplicative functions, and Ramanu-jan expansions.     

On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψk(x;q,a) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression . This theorem not only gives the asymptotic formula for ψk(x;q,a) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.     

On integers of the forms k−2 and k2+1

In this paper we prove that if (r,12)?3, then the set of positive odd integers k such that k^r−2ⁿ has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k^r2ⁿ+1 is also true.     

Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár

The Kalmár function K(n) counts the factorizations n=x₁x₂…xr with xi?2(1?i?r). Its Dirichlet series is where ζ(s) denotes the Riemann ζ function. Let ρ=1.728… be the root greater than 1 of the equation ζ(s)=2. Improving on preceding results of Kalmár, Hille, Erd?s, Evans, and Klazar and Luca, we show that there exist two constants C₅ and C₆ such that, for all n, holds, while, for infinitely many n's, .An integer N is called a K-champion number if M<N⇒K(M)<K(N). Several properties of K-champion numbers are given, mainly about the size of the exponents and the number of prime factors in the standard factorization into primes of a large enough K-champion number.The proof of these results is based on the asymptotic formula of K(n) given by Evans, and on the solution of a problem of optimization.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

On the average orders of the error term in the Dirichlet divisor problem

Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas and ∑_n?xΔ(n)^k with a natural number k. In particular we study the case k=2 and 3 in detail.     

On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m?2 and r with 0?r?m−1, let A(x;m,q,r) denote the numbers of positive integers n?x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).     

On the spacings between C-nomial coefficients

Let (Cn)_n?0 be the Lucas sequence Cn+2=aCn+1+bCn for all n?0, where C₀=0 and C₁=1. For 1?k?m−1 let

     

On the number of large oscillations of some arithmetical power series

We apply a method of J. Kaczorowski in order to study the number of large oscillations of the error term of weighted averages of some arithmetical functions. Received: 16 April 2002     

À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors

In this paper we consider an analogue of the problem of Erds and Woods for arithmetic progressions. A positive answer follows from theabc conjecture. Partial results are obtained unconditionally.     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2. Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland The author was supported in part by KBN Grant # N N201 1482 33.     

Pest control may make the pest population explode

We present an example of a predator-prey-like system with a prey-only state as a global attractor, and with the additional property that an attempt to control the prey by harvesting or poisoning both species produces solutions in which both populations blow up in finite time.