In this paper we investigate the solutions in integers x, y, z, X, Y, Z of the system

where P is a real number that may be taken to be arbitrarily large, and the (fixed) integer exponent h satisfies h≥4. The system has 6P3 + O(P2) “trivial” solutions in which x, y, z are a permutation of X, Y, Z. Our result implies that the number of non-trivial solutions is at most o(P3), so that the total number of solutions is asymptotic to 6P3.

Let (Sn)n = 1,2,… be the strictly increasing sequence of those natural numbers that can be represented as the sum of three cubes of positive integers. The estimate

is easily proved as follows: Let x1 be the largest natural number with Then This procedure is iterated by choosing x2 and then x3 as the largest natural numbers satisfying and Thus Since this implies (1).

Recently, it has been shown that tight or almost tight upper bounds for the discrepancy of many geometrically denned set systems can be derived from simple combinatorial parameters of these set systems. Namely, if the primal shatter function of a set system ℛ on an n-point set X is bounded by const. md, then the discrepancy disc (ℛ) = O(n(d−1)/2d) (which is known to be tight), and if the dual shatter function is bounded by const. md, then disc We prove that for d = 2, 3, the latter bound also cannot be improved in general. We also show that bounds on the shatter functions alone do not imply the average (L1)-discrepancy to be much smaller than the maximum discrepancy: this contrasts results of Beck and Chen for certain geometric cases. In the proof we give a construction of a certain asymptotically extremal bipartite graph, which may be of independent interest.

We generalize dual reductive pairs by using reductive groups that are not necessarily subgroups of symplectic groups and construct the corresponding theta-series liftings for certain types of automorphic forms. We also discuss connections of such generalized theta-series liftings with families of abelian varieties parametrized by an arithmetic variety.

An inhomogeneous version of a general form of the Jarník-Besicovitch Theorem is proved.

In 1984 Heath-Brown [5] proved the following conjecture of Erdős and Mirsky [2] (which seemed at one time as hard as the twin prime problem):

“There exist infinitely many integers n for which d(n) = d(n + 1).”

On Waring's problem for cubes, it is conjectured that every sufficiently large natural number can be represented as a sum of four cubes of natural numbers. Denoting by E(N) the number of the natural numbers up to N that cannot be written as a sum of four cubes, we may express the conjecture as E(N)≪1.

The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

In 1973, Montgomery [12] introduced, in order to study the vertical distribution of the zeros of the Riemann zeta function, the pair correlation function

where w(u) = 4/(4 + u2) and γjj = 1, 2, run over the imaginary part of the nontrivial zeros of ζ(s). It is easy to see that, for T → ∞,

uniformly in X, and Montgomery [12], see also Goldston-Montgomery [7], proved that under the Riemann Hypothesis (RH)

uniformly for X ≤ T ≤ XA, for any fixed A > 1. He also conjectured, under RH, that (1) holds uniformly for Xε ≤ T ≤ X, for every fixed ε > 0. We denote by MC the above conjecture.

We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Fröhlich.

The distribution of squarefree binomial coefficients. For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

Let Δn = {(x, y): x, y are integers 1 ≤ x, y ≤ n} be the n x n square array of integer lattice points in the plane.

Recently Bombieri and Sperber have jointly created a new construction for estimating exponential sums on quasiprojective varieties over finite fields. In this paper we apply their construction to estimate hybrid exponential sums on quasiprojective varieties over finite fields. In doing this we utilize a result of Aldolphson and Sperber concerning the degree of the L-function associated with a certain exponential sum.

This paper depends on results of Baranovskii [1], [2]. The covering radius R(L) of an n-dimensional lattice L is the radius of smallest balls with centres at points of L which cover the whole space spanned by L. R(L) is closely related to minimal vectors of classes of the quotient . The convex hull of all minimal vectors of a class Q is a Delaunay polytope P(Q) of dimension ≤, dimension of L. Let be a maximal squared radius of P(Q) of dimension n (of dimension less than n, respectively). If , then . This is the case in the well-known Barnes-Wall and Leech lattices. Otherwise, . This is a refinement of a result of Norton ([3], Ch. 22).

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.

Let K be a number field of degree nK = r1 + 2r2, a fixed integral ideal and the group of fractional ideals of K whose prime decomposition contains no prime factors of . Let

and be an arbitrary Groessencharaktere mod f as defined in [15]. Then and, for

where {λi} forms a basis for the torsion–free characters on whose value on any depends only on the that exists such that a = (α). Note that because of the choice in such an α we have that 1 for all units ε in K satisfying (mod ), ε>0. Also, x is a narrow ideal class character mod , that is, a character on

Let ((x)) = x −⌊x⌋−1/2 be the swatooth function. If a, b, c and e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

We study the Lw-norm (2 ≤ W < ∞) of the discrepancy of a sequence of points in the unit cube relative to similar copies of a given convex polygon. In particular, we prove the rather surprising result that the estimates obtained have the same order of magnitude as the analogous question when the sequence of points is replaced by a set of points.

The Redheffer matrix An = (aij)n×n defined by aij = 1 when i|j or j = 1 and aij = 0 otherwise has many interesting number theoretic properties. In this paper we give fairly precise estimates for its eigenvalues in punctured discs of small radius centred at 1.

The aim of the paper is to show the existence of a ‘Hall's ray’ for the particular case of the one-sided inhomogeneous diophantine approximation problem, where the irrational is the golden ratio. The proof uses a sum-set method similar to that used by Marshall Hall for the original result of this kind.