Let Ω(n) denote the number of prime factors of n, counted according to multiplicity. We shall consider the following question. Are there infinitely many natural numbers n for which Ω(n) = Ω(n + 1)? Erdős and Mirsky [4] have asked a closely related question concerning the divisor function d(n)—are there infinitely many n for which d(n) = d(n + 1)? The fact that Ω(n)is completely additive makes our problem slightly easier.

Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.

It is a familiar fact that the sequence of K-free numbers, i.e. those having no K-th power divisor greater than 1, has asymptotic density 1/ζ(K). Let χ(K) denote the characteristic function of this sequence, χ(2) = χ. Roth [5] proved that, if k/(n3/13(log n)4/13) → ∞, then

Before turning to the questions to be considered in this paper, we recall two other problems. Let C(a, p) be the class of all convex discs of area not less than a given constant a and perimeter not greater than a given constant p. What is the densest packing and what is the most economical covering of the Euclidean plane with discs from C(a, p)?

Both problems are interesting only if p2/a < 8√3, i.e. if p is less than the perimeter of a regular hexagon of area a. In this case, the densest packing arises from a regular hexagonal tiling by rounding off the corners of the tiles by equal circular arcs so as to obtain smooth hexagons of area a and perimeter p.

In this paper n always denotes an arbitrary but fixed positive integer. Let S be a subset of n-dimensional euclidean space En and (Si) = (S1, S2, …) a finite or infinite sequence of subsets of En. The sequence (Si) is called a covering of S if S ⊂ ⋃Si, and a packing in S if ⋃Si ⊂ S and int Si, ⋂ int Sj = Ø (for all i ≠ j). We say that (Si) permits an isometric covering of S or packing in S if there are rigid motions σi so that (σiSi) is a covering of S or a packing in S, respectively. If there are not only rigid motions but translations τi so that (τiSi) is a covering or packing, we express this by saying that (Si) permits a translative covering or packing. We consider sequences (Si) rather than sets {Si} not because the ordering is of any importance but because some of the sets Si may appear repeatedly.

Let be k non-overlapping translates of the unit d-ball Bd in euclidean d-space Ed. Let Ck denote the convex hull of their centres and let Sk be a segment of length 2(k– 1). Furthermore, let Vd denote the d-volume. L. Fejes Tóth conjectured in [1], that, for d ≥ 5,

The construction, for a module M over a commutative ring A (with identity) and a multiplicatively closed subset S of A, of the module of fractions S-1M is, of course, one of the most basic ideas in commutative algebra. The purpose of this note is to present a generalization which constructs, for a positive integer n and what is called a triangular subset U of An = A × A × … × A (n factors), a module U-n M of generalized fractions, a typical element of which has the form

where m∈ M and (u1, … un)∈U.

The classical mean value theorem for Dirichlet's polynomials states that

see H. L. Montgomery [7]. This formula is very useful in the theory of the Riemann zeta-function ζ(s). From the approximate functional equation

where | χ(½ + it)| = 1, u, v ≥ 1, 2πuv = t (see E. C. Titchmarsh [8]) it follows that χ(½ + it) can be well approximated by Dirichlet's polynomials of length N< t½.

This paper is concerned with two aspects of the theory of measures on compact totally ordered spaces (the topology is to be the order topology). In Section 2, we clarify a recent construction of Sapounakis [11, 12] and, in so doing, we are able to say a little more about it. It should be added here that Sapounakis had other ends in view. To be precise, let I be the closed unit interval [0, 1] and let λ be Lebesgue measure on I. We shall construct another totally ordered set Ĩ which is compact in its order topology, a continuous increasing surjection τ : Ĩ → I with the property that card τ−1(t) = 2 for all t ∈ ]0,1[ (these brackets denote the open interval), and a measure on Ĩ such that τ() = λ. Then the following theorem holds.

Let f be a non-zero cusp form of weight k on SL(2, ℤ) with Fourier expansion We assume further that f is normalized (a(1) = 1) and that f is an eigenfunction of the Hecke operators. Define

Let q = pn, p a rational prime, and let be the finite field with q elements. The polynomial ring is considered as an analogue of the ring of rational integers ℤ. Completing the quotient field with respect to the normalized valuation at ∞, and then taking algebraic closure, we obtained the field k∞ whose elements will be called “numbers”.

Hyperplane mean values of non-negative subharmonic functions have been studied in many papers, of which [2] and [3] are examples. Recently, Armitage [1] began a study of hyperplane means of non-negative superharmonic functions. One of his results [1, Theorem 2] shows that, if w is a positive superharmonic function on

and n/(n + 1) p < 1, then

as t → ∞, while another [1, Theorem 1] shows that, if 0 < p ≤ n/(n + l), then the integral in (1) is always infinite. However, he did not present a complete analogue of the result of Flett and Brawn [2, 3], which states that, if Ф: [0, ∞ [ → [0, ∞ [ is a non-decreasing convex function such that Ф(u)/u → 0 as u → 0 +, and w is a nonnegative subharmonic function on Rn × ]0, ∞ [, then under certain conditions on the size of w, the integral mean

tends to zero as t → ∞. In this note we present an analogue for superharmonic functions of the above result, in which the mean M(Ф(w); t) is shown to tend to infinity with t, provided that Ф(u)/u → ∞ as u → 0 +, and which therefore generalizes (1). It might be expected that, in dealing with the superharmonic case, the function Ф would have to be concave, so that Ф(w) would also be superharmonic. It turns out that this condition is unnecessary.

We consider the Falkner–Skan equation

in the special case β = – 1. It is known that this equation may be integrated twice to get the Riccati equation

Two spheres of different radii are approaching each other with equal and opposite velocities, the fluid flow around them being at low Reynolds number. The forces on the spheres can be calculated when they are very close by applying an asymptotic analysis — usually called lubrication theory — to the flow in the gap between the spheres. If the non-dimensional gap width is ε, the force is calculated here correct to O(ε In ε) for all ratios of the two spheres' radii. The analysis can be combined with earlier numerical calculations to find all the constants in the asymptotic expansion correct to O(ε).

If C and Co are two convex bodies in Ed we say that C slides (rolls) freely inside Co if the following condition is satisfied: for each x ∈ ∂C0 (and each rotation R) there is a translation t such that, if gC = C + t (= RC + t), then gC ⊂ Co and x ∈ ∂gC. This work establishes certain topological conditions which ensure the free rolling and sliding of C inside Co. One consequence of these conditions is that, if ∂K ∩ int gK is a topological ball for all rigid motions g, then K is a ball in the geometrical sense.

Given a second-order, linear, partial differential equation, it is sometimes the case that an arbitrary non-negative solution on a strip or half-space ℝn × ]0, c[, where 0 < c ≤ ∞, can be represented by the integral of a kernel function with respect to a non-negative measure on ℝn. The solution is thus, at least theoretically, determined by the measure. This paper is concerned with the determination of the measure, given the solution.

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

Let λ1, …, λ8 be any non-zero real numbers not all of the same sign and not all in rational ratios. According to a theorem of Davenport and Roth [3], given a real number κ, the inequality

has infinitely many solutions in positive integers for any ε > 0. Recently Liu, Ng and Tsang [5] gave a refinement of this result: for any δ > 0, the inequality

has infinitely many solutions in positive integers. In the present note we obtain a better exponent.

Let K be an algebraic number field of degree n = rl + 2r2 (in the usual notation) over the rationals with discriminant d. Let ZK denote the ring of integers in K. It is usual to speak of an integer Πi ∈ Zk as an almost-prime of order l, if the principal ideal (Πi) has at most l prime ideal factors, counted according to multiplicity. Let P1, …, Pn be positive real numbers with Pk = Pk+r2, k = r1 + l, …, r1 + r2 and P = P1 … Pn ≥ 1.

Denote by N(t) the number of zeros β + iγ of ζ(s), s = σ + it, with 0 < β < 1 and 0 < γ < t. It is well known that

where