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     

A Mean Value Theorem for Dirichlet Series and a General Divisor Problem

 We consider the mean value formula for general Dirichlet series by applying the approximate functional equation of Ramachandra’s type, and derive the sum formula for its coefficients. The improvement of Landau’s classical results is established for the general divisor function. We also obtain the asymptotic behaviour of the mean value when the real part of s is near the abscissa of absolute convergence. Received 2 July 2001; in revised form 29 October 2001     

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.     

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.     

On the Primitive Circle Problem

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

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory. Received October 18, 2001; in final form April 11, 2002     

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 values of the divisor function

For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2ⁿ − 1) : n ≤ x}. Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia, Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia     

On Exponential Sums Involving the Ideal Counting Function in Quadratic Number Fields

 Let χ be a Dirichlet character modulo k > 1, and F _χ(n) the arithmetical function which is generated by the product of the Riemann zeta-function and the Dirichlet L-function corresponding to χ in . In this paper we study the asymptotic behaviour of the exponential sums involving the arithmetical function F _χ(n). In particular, we study summation formulas for these exponential sums and mean square formulas for the error term. Received April 17, 2001; in revised form April 2, 2002     

On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1]. (Received 29 June 1998; in final form 11 November 1998)     

On the Value Distribution of Shifts of Universal Dirichlet Series

Shifts of universal Dirichlet series are jointly universal, that means, roughly speaking, they approximate simultaneously any given family of non-vanishing analytic functions. In this note we study consequences of this remarkable property on the value-distribution. Partially supported by the Foundation for Polish Science (FNP) and KBN grant 1 PO3A 00826. Partially supported by Lithuanian Foundation for Studies and Science.     

On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product

The First Power Mean of the Inversion of L-Functions and General Kloostermann Sums

 The main purpose of this paper is using the estimate for classical Kloostermann sums and E. Bombieri–A. I. Vinogradov’s important work to study the first power mean of the inversion of Dirichlet L-functions with the weight of general Kloostermann sums, and give an interesting asymptotic formula. Received 30 March 2001; in revised form 12 November 2001     

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 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.     

The k-free Divisor Problem

 Let be the number of k-free divisions of n, and let be the counting function of . We improve on the known estimates for the error term in the asymptotic formula for under the assumption of the Riemann Hypothesis. We also obtain an unconditional asymptotic formula for , for small y. (Received 15 March 1999; in revised form 17 September 1999)     

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 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 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