On Sums of Two Coprime k-th Powers(II)

For fixed k ≥ 3, let E_k(x) denote the error term of the sum , where 1. It is proved that if the Riemann hypothesis is true, then , . A short interval result is also obtained.     

On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ (𝒮(t))²dt for any Λ = Λ(T) growing faster than log T. Received April 30, 2001; in revised form February 15, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Almost Perfect Powers in Products of Consecutive Integers

The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k 5, under the assumption that P(b) p_k, the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.     

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.     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

Sums and Differences of Two Cubic Polynomials

 When f(x) is a cubic polynomial with integral coefficients, we show that almost all integers represented as the sum or difference of two values of f(x), with , are thus represented essentially uniquely. (Received 18 January 1999; in revised form 17 May 1999)     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

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     

Lattice Points in Planar Convex Domains

We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero.     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds. Received 25 April 2001     

Effektive Abschätzungen für den Gitterrest gewisser ebener und dreidimensionaler Bereiche

Effective estimates for the lattice point discrepancy of certain planar and three-dimensional domains. This paper provides estimates, with explicit constants, for the lattice point discrepancy of o-symmetric ellipse discs and ellipsoids in ℝ³, as well as of three-dimensional convex bodies which are invariant under rotations around one coordinate axis and have a smooth boundary of finite nonzero Gaussian curvature throughout.Der zweitgenannte Autor denkt für die Unterstützung durch den Österreichischen Forschungs förderungs fonds (FWF), Projekt Nr. P18079-N12.     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method. Received 25 August 2000; in revised form 8 January 2002     

A Lower Bound for the Height of a Rational Function at <Emphasis Type="Italic">S</Emphasis>-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

On Vectors whose Span Contains a Given Linear Subspace

Estimates are given for the product of the lengths of integer vectors spanning a given linear subspace.The first author was supported by FWF Austrian Science Fund, project M672.     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

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     

Sums of two k-th powers: an Omega estimate for the error term

The arithmetic function r _k (n) counts the number of ways to write a natural number n as a sum of two k-th powers (k ≧ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of r _k(n) leads in a natural way to a certain error term P _k(t). In this article, we establish an Ω-estimate for P _k(t) (k τ; 2 arbitrary) which is essentially as sharp as the best known one in the classic case k=2. This article is part of a research project supported by the Austrian Science Foundation (Nr. P 9892-PHY).     

On the Radial Behavior of Universal Taylor Series

In this work we deal with universal Taylor series in the open unit disk, in the sense of Nestoridis; see [12]. Such series are not (C,k) summable at every boundary point for every k; see [7], [11]. In the opposite direction, using approximation theorems of Arakeljan and Nersesjan we prove that universal Taylor series can be Abel summable at some points of the unit circle; these points can form any closed nowhere dense subset of the unit circle.     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.