Some Examples in the Theory of Subgroup Growth

By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.     

One-Sided Ideal Growth of Free Associative Algebras

Let R be a finitely generated associative algebra with unity over a finite field \Bbb F_q{\Bbb F}_q . Denote by a _n (R) the number of left ideals J ⊂ R such that dim R/J = n for all n ≥ 1. We explicitly compute and find asymptotics of the left ideal growth for the free associative algebra A _d of rank d with unity over \Bbb F_q{\Bbb F}_q , where d ≥ 1. This function yields a bound a _n (R) ≤ a _n (A _d ), n ? \Bbb Nn\in{\Bbb N} , where R is an arbitrary algebra generated by d elements. Denote by m _n (R) the number of maximal left ideals J ⊂ R such that dim R/J = n, for n ≥ 1. Let d ≥ 2, we prove that m _n (A _d ) ≈ a _n (A _d ) as n → ∞.     

Double Exponential Sums and Some Applications

.?We give three estimates on bilinear exponential sums of type I. As application we prove that , where the A _j are effective constants and t(?) denotes the number of unitary factors of a finite abelian group ?. This improves on the previous result.     

On the Primitive Circle Problem

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

Spectral synthesis on multivariate polynomial hypergroups

In a recent paper [16] we presented some results concerning spectral analysis and spectral synthesis on polynomial hypergroups in a single variable. Now we show that using Hilbert’s Nullstellensatz, the Noether-Lasker-theorem and the Ehrenpreis-Palamodov-theorem the ideas of [16] can be extended to prove spectral analysis and spectral synthesis on any multivariate polynomial hypergroup. Author’s address: Institute of Mathematics, University of Debrecen, Egyetem tér 1, 4032 Debrecen, Hungary     

Construction of pseudorandom binary lattices by using the multiplicative inverse

In an earlier work Hubert and the authors of this paper introduced and studied the notion of pseudorandomness of binary lattices. Later in another paper the authors gave a construction for a large family of “good” binary lattices by using the quadratic characters of finite fields. Here, a further large family of “good” binary lattices is constructed by using finite fields and the notion of multiplicative inverse. Authors’ addresses: Christian Mauduit, Institut de Mathématiques de Luminy, CNRS, UMR 6206, 163 avenue de Luminy, Case 907, F-13288 Marseille Cedex 9, France; András Sárk?zy, Department of Algebra and Number Theory, E?tv?s Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary     

An Additive Problem with Piatetski-Shapiro Primes and Almost-Primes

Suppose that . We prove a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes p = [n ^1/ and show that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.Received November 29, 2001; in revised form August 21, 2002 Published online October 15, 2003     

Explicit evaluations of Ramanujan-Göllnitz-Gordon continued fraction

In this paper, we establish several theorems for the explicit evaluations of Ramanujan-G?llnitz-Gordon continued fraction by using some parameterizations of Ramanujan’s theta-functions. Nayandeep Deka Baruah, Corresponding author. Research partially supported by grant SR/FTP/MA-02/2002 from DST, Govt. of India. Authors’ address: Nayandeep Deka Baruah and Nipen Saikia, Department of Mathematical Sciences, Tezpur University, Napaam-784028, Assam, India     

Degenerations of Representations of Associative Algebras

The purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finite-dimensional representations of finitely generated, and especially, finite-dimensional algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether. Received: July 2007     

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.     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form , where is a continuous function with , runs over , the set of Farey fractions of order Q in the unit interval [0,1] and are consecutive elements of . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer satisfying ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

Cubic Fields with Index 2

 Let d be a squarefree integer with allowed. If (mod 8) it is shown that there do not exist any cubic fields with index 2 whose splitting field contains . If (mod 8) it is shown that there exist infinitely many cubic fields with index 2 and minimal index 2 whose splitting field contains . (Received 23 May 2001)     

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.     

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.     

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     

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     

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     

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.     

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.