It is shown that in all dimensions n ≥ 11 there exists a lattice which is generated by its minimal vectors but in which no set of n minimal vectors forms a basis.

Let g(n) be a complex valued multiplicative function such that |g(n)| ≤ 1. In this paper we shall be concerned with the validity of the inequality

under the weak condition g(p)∈ for all primes p, where is a fixed subset of the closed unit disc Thus our point of view is similar to that of Halász [Hz 2] in that we seek a general inequality in terms of simple quantities, albeit g(p) may have a quite irregular distribution. We are not concerned here with the problem of asymptotic formulae for the sum on the left of (1) studied by (among others) Delange [D], Halász [Hz 1] and Wirsing [W].

The aim of this note is to give a sharp lower bound for rational approximations to ζ(2) = π2/6 by using a specific Beukers' integral. Indeed, we will show that π2 has an irrationality measure less than 6.3489, which improves the earlier result 7.325 announced by D. V. Chudnovsky and G. V. Chudnovsky.

Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation

in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula

whenever s≥(k−2)2k−1 + 5. Here , the singular series, relates the local solubility of (1.1). For each k we define to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.

We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.

We shall give an explicit form of the Artin-Tschebotareff density theorem in function fields with several variable over finite fields. It may be an analogous prime number theorem in the higher dimensional case.

We prove a result related to the Erdős-Ginzburg-Ziv theorem: Let p and q be primes, α a positive integer, and m∈{pα, pαq}. Then for any sequence of integers c= {c1, c2,…, cn} there are at least

subsequences of length m, whose terms add up to 0 modulo m (Theorem 8). We also show why it is unlikely that the result is true for any m not of the form pα or pαq (Theorem 9).

For any fixed positive real number ε, any integer b≥2 and any dε{0, 1,…, b−1}, the set of Borel's simply normal numbers to base b in [0, 1] is partitioned into a countable number of sets in eight different ways according to the largest place and the number of places at which the proportion d's to that place in the b-adic expansion of such a number exceeds or is not less than b−1 – ε, or is less than or does not exceed b−1 – ε. For selected values ε, the Lebesgue measures of the sets in these decompositions are given explicitly.

In this paper the Hausdorff dimension of systems of real linear forms which are simultaneously small for infinitely many integer vectors is determined. A system of real linear forms,

where ai, xij∈ℝ, 1 ≤i≤m, 1≤j≤n will be denoted more concisely as

where a∈⇝m, X∈ℝmn and ℝmn is identified with Mm × n(ℝ), the set of real m × n matrices. The supremum norm of any vector in k dimensional Euclidean space, ℝk will be denoted by |v|. The distance of a point a from a set B, will be denoted by dist (a, B) = inf {|a − b|: b ∈ B}.

In the case of F-isotropic groups for a global field F, Moore [Mo] computed the metaplectic kernel using crucially his theorem of uniqueness of reciprocity laws. For F-anisotropic G, a variant of Moore's theorem is, therefore, needed to compute the metaplectic kernel. Such a variant was announced by G. Prasad [GP1] (in 1986) and here we give the details.

Let k ≥ 3 and n > 6k be positive integers. The equations, with integer coefficients, have nontrivial p-adic solutions for all p > Ck8, where C is some positive constant. Further, for values k≥ K we can take C = 1 + O(K-½).

Let |θ| < π/2 and . By refining Selberg's method, we study the large values of as t → ∞ For σ close to ½ we obtain Ω+ estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that

and

Our results supplement those of Montgomery which are good when σ > ½ is fixed.

In Mahler's classification of complex numbers [10] (see [4]), a transcendental number ξ is called a U-number if there exists a fixed integer N ≥ 1 so that for all ω > 0, there exists a polynomial so that

where the height h(f) = max {|α0|, |α1|, …, |αN|}. The number ξ is called a Um-number if the above holds for N = m but for no smaller value of N (examples and further details may be found in [9,1 and 2]). Thus the set of U1-numbers is precisely the set of Liouville numbers. In this paper we investigate the statistical behavior of the partial quotients of real U-numbers, in particular, U2-numbers. In addition, we demonstrate the existence of a U2-number with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number. Related results involving badly approximate U2-numbers are also discussed.

Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,

where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sum

or the difference

of positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruences

must be solvable in α β є k, respectively. Moreover the condition

must hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.

Let 1 ≤ M ≤ N − 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the Mth compound body of K, (K)m, in Euclidean (MN)-space. The compound body (K)M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (MN)-space determined by K and M. The sets K and (K)M are related by a number of well-known inequalities due to Mahler.

Here we generalize this theory to the geometry of numbers over the adèle ring of a number field and prove theorems which compare an adelic set with its adelic compound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

Suppose that is a distribution of N points in the unit square U = [0, 1]2. For every measurable set B in U, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2

We obtain explicit lower bounds on the lattice packing densities δL of superballs G of quite a general nature, and we conjecture that as the dimension n approaches infinity, the bounds are asymptotically exact. If the conjecture were true, it would follow that the maximum lattice-packing density of the Iσ-ball is 2−n(1+σ(1)) for each σ in the interval 1 ≤ σ ≤ 2.

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

Let s1, s2, … denote the squarefree numbers in ascending order. In [1], Erdős showed that, if 0 ≤ γ ≤ 2, then

where B(γ) is a function only of γ. In 1973 Hooley [4] improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for

in contrast to the range

derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.

Let

and A1, A2, …, An, a1, a2, …, an be positive integers. We consider the related integrals

and