A sequence {an} of integers is said to be primitive if whenever i ≠ j. For example, if n is any positive integer, the sequence

is primitive. This is an important example in the light of the elementary result (see [2; p. 244]) that if 0 < a1 < a2 < … < ar ≤ 2n is primitive then necessarily r ≤ n; i.e. at most half of the positive integers ≤ 2n can be members of the sequence. Besicovitch [2; p. 257] has obtained the surprising result that, given ε > 0, there exists an infinite primitive sequence {ai} such that

where A(n) denotes the number of ai ≤ n.

Let p be an odd prime, let q be a divisor of p – 1, and let α be a primitive q-th root modulo p. For each natural number r the metacyclic group Gr is defined by

Erdoső has asked whether there exists a positive constant c < 1 such that

where pn is the n-th prime. In 1943 Selberg «3» proved, on the assumption of the Riemann Hypothesis, that

uniformly for H > 0. By an elementary extension of Selberg's method we prove the following two results free of any unproved assumptions.

We begin by introducing some notations which will be used throughout the paper. Let denote a sequence of distinct positive integers. Let B denote a measurable subset of [0, 1) and {x} the fractional part of x. We write

so that f has period 1 on the real line; and we write

for the number of times {mkx} falls in B, minus the expected value of this number (m(B) ≤ Lebesgue measure of B).

By a zonotope we mean any set in Euclidean n-dimensional space Rn which can be written as a Minkowski (vector) sum of a finite number of line segments. A zonotope is a convex centrally-symmetric polytope, and all its faces are zonotopes. Familiar examples of three-dimensional zonotopes include the cube, rhombic dodecahedron, elongated dodecahedron (Figure 1) and truncated octahedron. Photographs of models of more complicated examples appear in [1, Plate II].

Let f(m;n) denote the largest integer so that, given any m integers a1 < … < am in [1, 2n], one can always choose f integers b1 < … < bf from [1, n], so that bi + bj = a1 (1 ≤ i ≤ j ≤ f; l ≤ l ≤ m) will never hold. Trivially f(m; n) ≥ n/ (m + 1). In this paper we shall attempt to improve upon this trivial bound by exploiting the possible irregularities of distribution of the sequence among certain congruence classes. One of our main results is

provided m ≥ log n. Related questions and results are also discussed.

1. W. Blaschke's kinematic formula in the integral geometry of Euclidean n-dimensional space gives a weighted measure to the set of positions in which a mobile figure K1 overlaps a fixed figure K0. In the simplest case, K0 and K1 are compact convex sets and all positions are equally weighted; we give this in more detail. Let Wq denote the q-th Quermassintegral of K1: Steiner's formula for the volume V of the vector sum K1 + λB of K1 and a ball of radius λ defines these set functions by the equation

see [4; p. 214]. Blaschke's formula [4; p. 243] gives

as the measure, to within a normalization, of overlapping positions of K1 relative to K0.

For some integer modulus q let F be a subset of Z(q), the additive group of residues modulo q. As in «1» we shall write

and

The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.

In this paper I prove the following result:

Theorem. Let f(d) be a multiplicative function such that |f(d)| ≤ 1 and ∑{ l/p : p ε P} = ∞, where P denotes the set of primes p for which f{p) = −1, and let v(n) denote the number of distinct prime factors ofn. Then for almost all n,

where A is an arbitrary constant > 3/e.

In this paper it is shown that a simplicial action of Zp (p a prime) on an n-dimensional simplicial complex which is a Poincaré duality space of formal dimension n for Zp coefficients cannot have just one isolated fixed point.

§1. Introduction and Summary. Throughout X is a complete separable metric space. We write K1 for the family of non-empty compact subsets of X. K1 may be endowed with a metric (first introduced by Hausdorff) under which K1 is complete and separable. We shall make use of the subbase for this metrizable topology of K1 given by sets of the two forms

for U open in X (see Kuratowski [4] or E. Michael [9] for a discussion of topologies on the space of subsets of X). if we shall be concerned with sets in [0, 1] × X which are universal for ℋ. To define these let us make the convention that, for D ⊆ [0, 1] × X, we write

D is said to be universal for ℋ if

The stability against small disturbances which do not bend the vortex lines of a circular cylindrical vortex sheet of variable radius is examined. If the jump in tangential velocity across the sheet falls off with time faster than (time)–íthe disturbances are bounded (or grow less rapidly than linearly with time) so that the sheet is effectively stable. If the tangential velocity jump falls off with time slower than (time)–í the sheet is unstable.

The equations governing the evolution of the modulated amplitude of a pointcentred disturbance to a slightly supercritical flow are shown to have solutions which become infinite at a finite time and at a single point. The amplitude develops a sharp peak and the structure of this peak is found, for real and complex coefficients in the governing equations. Such a solution can only occur if the coefficients satisfy certain conditions

Using toroidal co-ordinates, an exact solution is derived for the velocity field induced in two immiscible semi-infinite viscous fluids, possessing a plane interface, by the slow rotation of a concave spherical lens, which is such that the circle of intersection of the composite spherical surfaces lies in the plane of the interface. The expression for the torque acting on the lens is derived, and this is shown to be reducible to an analytic closed form, when the lens degenerates into a spherical bowl and the fluids are identical.