For any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

One way of realizing representations of the Heisenberg group is by using Fock representations, whose representation spaces are Hilbert spaces of functions on complex vector space with inner products associated to points on a Siegel upper half space. We generalize such Fock representations using inner products associated to points on a Hermitian symmetric domain that is mapped into a Seigel upper half space by an equivariant holomorphic map. The representations of the Heisenberg group are then given by an automorphy factor associated to a Kuga fiber variety. We introduce theta functions associated to an equivariant holomorphic map and study connections between such generalized theta functions and Fock representations described above. Furthermore, we discuss Jacobi forms on Hermitian symmetric domains in connection with twisted torus bundles over symmetric spaces.

It is shown that a twist of a Dirichlet series by a Dirichlet character may lead to very different analytic properties.

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3 with Markoff triples.

The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

Existence criteria are presented for non-linear boundary value problems on the half line. In particular, the theory includes a problem in the theory of colloids and a problem arising in the unsteady flow of a gas through a semi-infinite porous medium.

A positive definite integral quadratic form over rational integers is said to be universal, if it represents all positive integers. The universal quaternary quadratic form is determined with the maximal discriminant, which is 1073/4.

An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.

Erdős (see [4]) asked if there are infinitely many integers k which are not a difference or a sum of two powers, i.e., if there are infinitely many positive integers k with k≠|um ± vn| for u, v, m, n ε ℤ. This is certainly a very difficult problem. For example, it is known that the Catalan equation, i.e., the equation um − vn = 1 with uv≠0 and min (m, n)≥2 has only finitely many solutions (u, v, m, n), but there is no other positive integer k≥1 for which it is known that the equation

has only finitely many solution (u, v, m, n) with min (m, n)≥2.

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a2 + b2 counted in the usual way, that is, with regard to both the order and sign of a and b. We have

where x is the non-principal Dirichlet character (mod 4); moreover, r(n)≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b(n) on the sequence of representable numbers as the least possible value of |b|—for example, we have b(13) = 2,b(25) = 0, b(65) = 1—and we write

here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x).

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

For an indefinite quadratic form f(x1,…,xn) of discriminant d. let P(f) denote the greatest lower bound of the positive values assumed by f for integers x1, …, xn. This paper investigates the values of P3/|d| for nonzero ternary forms of signature −1, and finds the only remaining class of forms with .

Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above is

for 0<l<½ and with the notation that, for any α ∈ ℝ, we set β. where β is the unique real satisyfing .

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

General upper bounds for lattice kissing numbers are derived using Hurwitz zeta functions and new inequalities for Mellin transforms.

We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function

Improved upper and lower bounds of the counting functions of the conceivable additive decomposition sets of the set of primes are established. Suppose that where, ℝ′ differs from the set of primes in finitely many elements only and .

It is shown that the counting functions A(x) of ℐ and B(x) of ℬ for sufficiently large x, satisfy

The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more generally, prime ideals in algebraic number fields). One consequence of our main theorem (Theorem 2 below) is the following result on rational primes.

Let (bn) be a sequence of integers, obtained by traversing the rows of Pascal's triangle, as follows: start from the element at the top of the triangle, and at each stage continue from the current element to one of the elements at the next row, either the one immediately to the left of the current element or the one immediately to its right. Consider the distribution of the sequence (bnα) modulo 1 for an irrational α. The main results show that this sequence “often” fails to be uniformly distributed modulo 1, and provide answers to some questions raised by Adams and Petersen.