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.

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 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,

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

This paper contains estimates of several exponential sums involving Fourier coefficients of certain modular forms. Although the questions which we consider date back, in some cases, to the twenties, our motivation is modern. All of these sums are connected with one approach to the problem of calculating the norm of the Poincaré 0-operator. In this section we state our results (and their antecedents) and then describe their relevance to the θ-operator problem. For simplicity we state and prove all our results for the modular cusp form of weight six, the famous discriminant function

The purpose of this paper is to provide additional evidence to support our view that the modules of generalized fractions introduced in [8] are worth further investigation: we show that, for a module M over a (commutative, Noetherian) local ring A (with identity) having maximal ideal m and dimension n, the n-th local cohomology module may be viewed as a module of generalized fractions of M with respect to a certain triangular subset of An + 1, and we use this work to formulate Hochster's ‘Monomial Conjecture’ [2, Conjecture 1]; in terms of modules of generalized fractions and to make a quick deduction of one of Hochster's results which supports that conjecture.

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 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.

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1F) = m(F) for all F ∈ A and all surjective group endomorphisms T of G onto G.

The aim of this paper is to give a clear statement, and, I hope, a reasonably clear proof, of a theorem of Thorn, which occurs in his important and difficult paper “Ensembles et morphismes stratifiés” [10]. The theorem to which I refer is Théorème 1.D.1 of [10]. “Tout espace stratifié compact admet une présentation associée aux applications kYX données”. At least, I think that the theorem herein described is equivalent to the above, but I could not swear to it. The main difficulty is that, despite strenuous efforts on my part, I have always found it easier to rig up my own system of definitions than to work within the framework suggested by Thorn. However, the two accounts clearly say the same sort of thing. In particular, §1 of the present paper is closely related to, and heavily influenced by, the material on page 250 of [10].

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.

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.

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.

Let ℱ denote a set of subsets of X = {1, 2,…, n). Let deg(i) be the number of members of ℱ containing i and val(ℱ) = min {deg (i): i ∈ X). Suppose no k members of ℱ have union X. We conjecture val(ℱ) ≤ 2n-k-1 for k ≥ 3. This is known for n ≤ 2k and we prove it for k ≥ 25. For k = 2 an example has val(ℱ) > 2n-2(l−n-0·651) and we prove val(ℱ) ≤ 2n-2(1–n-1). We also prove that if the union of k sets one from each of ℱ1,…, ℱk has cardinality at most n – t then min {cardinality ℱj} < 2nαt where αk = 2α − 1 and ½ < α < 1.

If П is a k-dimensional vector subspace of Rn and E is a subset of Rn, let projп(E) denote the orthogonal projection of E onto П. Marstrand [8] and Kaufman [6] have developed results on the Hausdorff dimension and measure of projп(E) in terms of the dimension of E, leading to the very general theory of Mattila [11]. In particular, Mattila shows that if the Hausdorff dimension dim E of the Souslin set E is greater than k, then projп(E) has positive k-dimensional Lebesgue measure for almost all П ∈ Gn, k (in the sense of the usual normalized invariant measure on the Grassmann manifold Gn, k of k-dimensional subspaces of Rn).

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.

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.

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.