In 1918 Pólya and Vinogradov established the estimate for Dirichlet character sums that currently carries their names. It was forty years until Burgess gave an improvement of their bound [1], and it is forty years since that improvement.

The question as to which “natural” sequences contain infinitely many primes is of considerable fascination to the number-theorist. One such “natura” sequence is [nc[ where [·] denotes integer part. Piatetski-Shapiro [10] showed that there are infinitely many primes in this sequence for 1 < c < 12/11, obtaining the expected asymptotic formula for the number of such primes. The exponent 12/11 has been increased gradually by a number of authors to the present record 45/38 held by Kumchev [9]. It is expected that there are infinitely many primes of the form [nc[ for all cεε[1, ∞)/ℤ. Deshouillers [3] showed that this is almost always true, in the sense of Lebesgue measure on [1, ∞). Balog [2] improved and generalized this result to show that, for almost all c > 1,

where

As usual in Waring's problem, we let G(k) be the least number s such that all sufficiently large natural numbers can be written as the sum of s or fewer k-th powers of positive integers. Hardy and Littlewood gave the first general upper bound, G(k)≤(k − 2)2k − 1 + 5. Later [4], they reduced this to order k2 by first constructing an auxiliary set of natural numbers below x which are sums of sk-th powers, of cardinality with for large k. The argument, which is brief and elementary (see [15, Chapter 6]), is now referred to by R. C. Vaughan's term “diminishing ranges”, since the k-th powers are taken from intervals of decreasing lengths. This idea of choosing the variables from restricted ranges was refined by Vinogradov, whose application to exponential sum estimates gave for large k (see [18]). The recent iterative method of Vaughan and Wooley [16, 19, 21], which halves Vinogradov's bound, may be viewed as an evolved diminishing ranges argument, producing an auxiliary set with .

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k2r + 2. In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r2k2+(2/(c − 2)) if r ≠ 1 and p > k2+(2/(c − 1)) if r = 1.

Recent advances in the theory of exponential sums (see, for example, [6], [7], [8], [12]) have contributed to corresponding progress in our understanding of the solubility of systems of simultaneous additive equations (see, in particular, [1], [2], [3], [4]). In a previous memoir [11] we developed a version of Vaughan's iterative method (see Vaughan [8]) suitable for the analysis of simultaneous additive equations of differing degrees, discussing in detail the solubility of simultaneous cubic and quadratic equations. The mean value estimates derived in [11] are, unfortunately, weaker than might be hoped, owing to the presence of undesirable singular solutions in certain auxiliary systems of congruences. The methods of [12] provide a flexible alternative to Vaughan's iterative method, and, as was apparent even at the time of their initial development at the opening of the present decade, such ideas provide a means of avoiding altogether the aforementioned problematic singular solutions. The systematic development of such an approach having been described recently in [15], in this paper we apply such methods to investigate the solubility of pairs of additive equations, one cubic and one quadratic, thereby improving the main conclusion of [11].

For a two parameter family of C3 diffeomorphisms having a homoclinic orbit of tangency derived from a horseshoe, the relationship between the measure of the parameter values at which the diffeomorphism (restricted to a certain compact invariant set containing the horseshoe) is not expansive and the Hausdorff dimension of the horseshoe associated to the homoclinic orbit of tangency is investigated. This is a simple application of the Newhouse-Palis-Takens-Yoccoz theory.

This is an expanded version of two lectures given at the conference held at Sydney University in December 1997 on the 50th anniversary of the death of G. H. Hardy.

Suppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators

L'objet de cet article est d'étudier un procédé de summation associé á certaines séries. Notant P(n) le plus grand facteur premier d'un entier générique n, nous rappellons les définitions de P-convergence et de P-régularité d'une série, introduites dans [7].

Let K be a nonarchimedean local field, let L be a separable quadratic extension of K, and let h denote a nondegenerate sesquilinear formk on L3. The Bruhat-Tits building associated with SU3(h) is a tree. This is applied to the study of certain groups acting simply transitively on vertices of the building associated with SL(3, F), F = Q3 or F3((X)).

A recurring theme in number theory is that multiplicative and additive properties of integers are more or less independent of each other, the classical result in this vein being Dirichlet's theorem on primes in arithmetic progressions. Since the set of primitive roots to a given modulus is a union of arithmetic progressions, it is natural to study the distribution of prime primitive roots. Results concerning upper bounds for the least prime primitive root to a given modulus q, which we denote by g*(q), have hitherto been of three types. There are conditional bounds: assuming the Generalized Riemann Hypothesis, Shoup [11] has shown that

where ω(n) denotes the number of distinct prime factors of n. There are also upper bounds that hold for almost all moduli q. For instance, one can show [9] that for all but O(Y∈) primes up to Y, we have

for some positive constant C(∈). Finally, one can apply a much stronger result, a uniform upper bound for the least prime in a single arithmetic progression. The best uniform result of this type, due to Heath-Brown [7], implies that . However, there is not at present any stronger unconditional upper bound for g*(q) that holds uniformly for all moduli q. The purpose of this paper is to provide such an upper bound, at least for primitive roots that are “almost prime”.

The following theorem, due to Artin [1], is one of a number of statements often referred to as Chebotarev density theorem.

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

The theory of isogeny estimates for Abelian varieties provides ‘additive bounds’ of the form ‘d is at most B’ for the degrees d of certain isogenies. We investigate whether these can be improved to ‘multiplicative bounds’ of the form ‘d divides B’. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes ℒ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for ℒ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).

It has been known that mixed automorphic forms arise naturally as holomorphic forms on elliptic varieties and that they include classical automorphic forms as a special case. In this paper, we show how to construct mixed automorphic forms of type (k, l) from elliptic modular forms to give nontrivial examples of mixed automorphic forms.

In [CKR], Chan, Kim and Raghavan determine all universal positive ternary integral quadratic forms over real quadratic number fields. In this context, universal means that the form represents all totally positive elements of the ring of integers of the underlying field. This generalizes the usage of the term introduced by Dickson for the case of the ring of rational integers [D]. In the present paper, we will continue the investigation of quadratic forms with this property, considering positive quaternary forms over totally real number fields. The main goal of the paper is to prove that if E is a totally real number field of odd degree over the field of rational numbers, then there are at most finitely many inequivalent universal positive quaternary quadratic forms over the ring of integers of E. In fact, the stronger result will be proved that this finiteness holds for those forms which represent all totally positive multiples of any fixed totally positive integer. The necessity of the assumption of oddness of the degree of the extension for a general result of this type can be seen from the existence of universal ternary forms over certain real quadratic fields (for example, the sum of three squares over the field ℚ(√5), as first shown by Maass [M]).

For a fixed integer k ≥ 2 suppose we have functions Li (x) satisfying the following conditions.

Crittenden and Vanden Eynden conjectured that if n arithmetic progressions, each having modulus at least k, include all the integers from 1 to k2n-k+1, then they include all the integers. They proved this for the cases k = 1 and k = 2. We give various necessary conditions for a counterexample to the conjecture; in particular we show that if a counterexample exists for some value of k, then one exists for that k and a value of n less than an explicit function of k.

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

By a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.