The exponential sum S(x) = Σe(f(m + x)) has mean square size O(M), when m runs through M consecutive integers, f(x) satisfies bounds on the second and third derivatives, and x runs from 0 to 1.

In the paper [2] Hsia noted that the forms x2+xy+y2+9z2 and x2+3y2+3yz+3z2 constitute a genus and that both forms are regular; he asked whether there exist any other genera containing two or more regular forms. In this note it is proved that the forms

are regular. They constitute a genus with discriminant 27 (in the normalization used by Brandt and Intrau in [1]). It is noteworthy that Hsia's genus has the same discriminant.

For arbitrary f: R → R and ϒ ⊂ Z × R we define the set of quantized observations of f relative to ϒ as follows: for each integer n and each y∈R we write

(the supremum of an empty set is taken to be −∞ ) and we put

Thus for example and , where [x] (without subscript) denotes as usual the integer part of x.

Roth's Theorem says that given ρ < 2 and an algebraic number α, all but finitely many rational numbers x/y satisfy |α - (x/y)|< |y|-ρ. We give upper bounds for the number of these exceptional rationals when 3 ≤ ρ ≤ d, where d is the degree of α. Our result suplements bounds given by Bombieri and Van der Poorten when 2 > ρ ≤ 3; naturally the bounds become smaller as ρ increases.

We prove a generalization of a theorem of Ryshkov relating the Voronoï vectors of lattices to the defining conditions for the Minkowski fundamental domain . This is then used to prove that a Minkowski reduced basis of a lattice of dimension n < 7 consists of strict Voronoï vectors.

In this note we introduce recurring-with-carry sequences which generalize add-with-carry and subtract-with-borrow sequences introduced in [1], and describe periods of admissible recurring-with-carry sequences which include add-with-carry and subtract-with-borrow sequences with a few exceptions.

Let λ1, …, λs be nonzero real numbers and suppose that λ1/λs is irrational. In 1955, Davenport and Roth showed [6] that the values taken by

at integer points (x1, …, xs) are dense on the real line, provided that s≥8. In the present paper we obtain the same result with seven variables.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

For a class of functions containing polynomials over ℤm, we give an inequality relating the cardinality of the value set to the additive order of differences of elements in that set. To do this, we find some inequalities concerning the combinatorics of substrings of sequences on finite sets which are related to an interesting matrix inequality.

Let R⊂S be two orders in a number field, and let ER and ES be the respective groups of units in each ring. Then ES/ER and S/R are both finite. We consider the problem of bounding the order of ES/ER in terms of the index of R in S. In this paper we solve this problem in the special case that S/R is cyclic as a module over Z.

We study polynomials over an integral domain R which, for infinitely many prime ideals P, induce a permutation of R/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field of R. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

This article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

In this short paper, we shall give a new estimate for the exponential sum S(H, M, N), where

e( ξ,) = exp (2πiξ;) for a real number ξ, am and bn are complex numbers with |am| ≤ 1 and |bn| ≤ l, H, M, N ≤1, , x is a large number, ε is a sufficiently small positive number, and Y ≤ x(½)−ε (h ∼ H means 1≤h/H < 2 and so on). In making application of the Rosser-Iwaniec linear sieve of Iwaniec [6] to find almost primes in short intervals of the type (x − y, x], Halberstam, Heath-Brown and Richert [4] first considered an estimate for S(H, M, N) to the effect that

with MN as large as possible. Later, better estimates were given in Iwaniec and Laborde [7], and Fouvry and Iwaniec [3]. Of course, the most interesting case would be finding P2 numbers in a short interval (x − y, x]. The related estimate of [7] implies that (1) holds provided that

Let r, k, h be positive integers. For any Dirichlet character χ modulo k write

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

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.