Let S(x) be a trigonometric polynomial,

where N > 0 and M are integers, the an are arbitrary complex numbers, and e(x) = e2πix. In its basic form, the large sieve of Linnik and Rényi is an inequality of the form

Among several interesting analogues of Chebyshev's problem about the largest prime factor of

there is the question of the largest prime factor of

where a is a given non-zero integer and the product is taken over positive primes p. The latter subject appears to have been first treated by Goldfeld [2] and Motohashi [6], who showed that, if Px be the greatest prime factor in question, then there exists a constant such that Px > xθ for all sufficiently large x. Their method, which involved the use of both Bombieri's theorem and the Brun-Titchmarsh theorem, had some affinity with the earlier treatments of Chebyshev's original problem.

Let f(n) be a real-valued additive arithmetic function. For each positive rational integer n set

For the duration of this paper we shall assume that as n → ∞ both

Let Ω be an algebraic number field, and let NΩ ⊂ ℚ be the group of norms of fractional ideals of Ω. Then NΩ is a subgroup of the positive rationals; the latter is the direct sum of a denumerable infinity of infinite cyclic groups, and so it is free abelian; thus NΩ is free abelian, and, since it is not finitely generated, we must have qua abstract groups. The purpose of this paper is, in the first place, to find a “metrical ” way of distinguishing these isomorphic groups, and, to this end, we introduce the notion of Farey density, defined as follows; let X be a positive integer, and consider the Farey section ℱ(X) of order X, thus the set of all reduced positive fractions with denominator < X; then the quotient

measures the proportion of elements of ℱ(X) which are in NΩ, and, as X → ∞, it gives a measure of the “density ” of fractional ideal norms in the rational interval (0, 1).

In this paper we show that, for the Apollonian or osculatory packing C0 of a curvilinear triangle T, the dimension d(C0, T) of the residual set is equal to the exponent of the packing e(Co, T) = S. Since we have [5, 6] exhibited constructible sequences λ(K) and μ(K) such that λ(K) < S < μ(K), and μ(K)–λ(K) → 0 as κ → 0, we have thus effectively determined d(C0, T). In practical terms it is thus now known that 1·300197 < d(C0, T) < 1·314534.

Using toroidal coordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite fluids possessing a plane interface, by the slow rotation of an axially symmetric body partly immersed in each fluid. The surface of the body is assumed to be formed from two intersecting spheres, or a sphere and a circular disc, with the circle of intersection of the composite surfaces lying in th interface.

It is shown that when the rotating body possesses reflection symmetry about the plane of the interface of the fluids, the velocity field in either fluid is independent of the viscosities of the fluids. The torque exerted on the body is then proportional to the sum of the viscosities. Analytic closed-form expressions are derived for the torque when the body is either a sphere, a circular disc, or a tangent-sphere dumbbell, and for a hemisphere rotating in an infinite homogeneous fluid. Closed-form results are also given for an immersed sphere, tangent to a free surface. For other geometrical configurations, numerical values of the torque are provided for a variety of body shapes and two-fluid systems of various viscosity ratios.

The Picone identity which occurs in the oscillation theory of second-order ordinary differential equations is

Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K(G) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ(G) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph Kp we have

A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation

The notion of nonatomicity of a measure on a Boolean σ-algebra is an important concept in measure theory. What could be an appropriate analogue of this notion for charges defined on Boolean algebras is one of the topics dealt with in this paper. Analogous to the decomposition of a measure on a Boolean σ-algebra into atomic and nonatomic parts, no decomposition of charges is available in the literature. We provide a simple proof of such a decomposition. Next, we study the conditions under which a Boolean algebra admits certain types of charges. These conditions lead us to give a characterisation of superatomic Boolean algebras. Babiker' [1] almost discrete spaces are connected with superatomic Boolean algebras and a generalisation of one of his theorems is obtained. A counterexample is also provided to disprove one of his theorems. Finally, denseness problems of certain types of charges are studied.

Our result complements an interesting result of Roy O. Davies [1]; we assume familiarity with his paper. We use the details of the construction that he uses to prove his Theorem II.

The letters a, b, n, m, t (perhaps with suffixes) always denote natural numbers. A, B, S denote finite sets of natural numbers. |A| stands for the cardinality of A.

For a given constant c > 1 we say that the set A has property α(c) if there are at most c|A| differences a − b ≥ 0 for a, b ∈ A.

Let pn denote the n-th prime number, and let

It follows from the prime number theorem that E ≤ 1. Erdős used a sieve argument to show that E < 1, and Rankin and Ricci gave the explicit estimates E ≤ 57/59 and E ≤ 15/16 respectively. A more powerful approach used by Hardy and Little-wood and by Rankin depended on hypotheses about the zeros of L-functions until Bombieri's Theorem was found. With its aid Bombieri and Davenport [1] proved that

A fuller history, with bibliography, is to be found in [1]

The purpose of this article is to give the proof of a theorem announced in [1]. We refer the reader to [1] for the terminology used here.

In [2], Sawyer considers a closed, central, convex region K which is such that, however it is displaced in the plane, a point of the integral lattice is covered. He shows that the area A(K) of K satisfies . We prove here a result in the opposite direction.

Our investigation starts from the question: is the even dimensional cohomology of the non-abelian group of order p3 and exponent p generated by Chern classes? From the computation of the complete cohomology ring in [8] one quickly sees that the essential problem is to express elements of the form cor (γk), γ ∈ H2 (k, ℤ), K a subgroup of index p, in terms of Chern classes. For a more general pair of groups (K ≤ G) it is known, see [7] that the best for which one can hope is a description of some multiple of cor (γk) in this way. Our first theorem shows that, under suitable hypotheses (satisfied in particular by the example of order p3) the numerical factor may be removed. Thus

In this paper is proved a formula for the number of simplicial neighbourly d-polytopes with d + 3 vertices, when d is odd.