On the Mean Value of the Product of Multiplicative Functions with Shifted Argument

In this paper, we consider the mean value of the product of multiplicative arithmetic functions with shifted argument. The investigated functions have to satisfy the following conditions: their moduli do not exceed 1; the values on the set of primes are close to 1 for one of the functions and close to a fixed complex number for the other function. Some consequences for the classical functions are given.     

Short Interval Results for a Class of Integers

For any subset , we introduce the definition of -numbers, which contains the well-known k-free numbers, k-full numbers, k-full and l-free numbers (k<l–1) as special cases. In this paper we study the distribution of -numbers in short intervals. We establish the connection of this problem with the gap problem of k-free numbers and multi-dimensional divisor problems. As applications, we study the short interval distribution of k-full and l-free numbers for k=2, 3, 4, 5, 6, 7.Received June 4, 2002; in revised form January 22, 2003 Published online June 30, 2003     

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.     

On the Primitive Circle Problem

 We prove that under the Riemann hypothesis one has for any ,

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,

Summatory Formula of the Convolution of Two Arithmetical Functions

 We study the asymptotic formula of for some arithmetical functions f and g. This generalizes the case investigated by Balakrishnan and Pétermann. Received 15 January 2001; in revised form 7 July 2001     

Oscillations of Error Terms Associated with Certain Arithmetical Functions

We prove a general theorem on oscillatory properties of error terms associated with real arithmetical functions whose Mellin transform may have singularities with several summands containing arbitrary complex powers and logarithmic polynomials. The theorem generalizes a theorem of J. Kaczorowski and J. Pintz and is motivated by the applications to algebraic integers with factorizations of distinct lengths. An application of the theorem to the study of the Hilbert semigroup modulo 5 is presented.     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria     

On Linear Independence of Theta Values

We prove a linear independence result for the values of theta series and Tschakaloff functions T_q(z) with different values of q.The first named author was supported in part by Grant-in-Aid for Scientific Research (No. 13640007), The Ministry of Education, Science, Sports, Culture of Japan.     

Discrepancy of Fractions with Divisibility Constraints

We provide bounds for the absolute discrepancy of sequences of fractions with denominators streaming in a given arithmetic progression and satisfying divisibility constraints. Supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

On some equations in prime rings

The main purpose of this paper is to prove the following result. Let R be a prime ring of characteristic different from two and let T : R → R be an additive mapping satisfying the relation T(x ³) = T(x)x ² − xT(x)x + x ² T(x) for all x ∈ R. In this case T is of the form 4T(x) = qx + xq, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.     

On abelian quantum invariants of links in 3-manifolds

From a finite abelian group G, a quadratic form onG and an element in , we define a topological invariant of a pair(M,L) where is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin-structures and related structures which appear in this context. Received May 13, 1998 / Accepted November 11, 1999 / Published online February 5, 2001     

On Some Exponential Diophantine Equations

 We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000)     

On a Problem of Erdős Involving the Largest Prime Factor of <Emphasis Type="Italic">n</Emphasis>

Let P(n) denote the largest prime factor of an integer n (2), and let N (x) denote the number of natural numbers n such that 2 n x, and n does not divide P(n)!. We prove that

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality is solvable, where p ₁,···,p ₅ are primes.     

Metric properties and exceptional sets of β-expansions over formal Laurent series

This paper is concerned with the metric properties of β-expansions over the field of formal Laurent series. We will see that there are essential differences between β-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined. Authors’ addresses: Bing Li and Jian Xu, School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R. China; Jun Wu, Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China     

Operator Semi-Selfdecomposable Measures and Related Nested Subclasses of Measures on Vector Spaces

T-semi-selfdecomposability, operator selfdecomposability, and subclasses L_m(b, (T_t)) and of measures on complete separable metric vector spaces are introduced and basic properties are proved. In particular, we show that is T-semi-selfdecomposable if and only if =T() where is infinitely divisible and is operator selfdecomposable if and only if L₀(b, (T_t)) for all 0<b<.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

The Best Constant in Siegel’s Lemma

We give a new proof of the basis form of Siegels Lemma over an algebraic number field k in which the field and dimension dependent constant is best possible. This constant is equal to a generalization of Hermites constant for the algebraic number field k that has recently been studied by J. L. Thunder.Research supported in part by the National Science Foundation (DMS-00-88915).Communicated by W. SchmidtReceived April 4, 2002; in revised form April 28, 2003 Published online August 28, 2003