The celebrated theorem of Bombieri and A. I. Vinogradov states that

for any ε > 0 and A ε 0, the implied constant in the symbol «ε depending at most on ε and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applications of sieve methods. Having this in mind distinct generalizations have been investigated (see for example [15] and [2]). Y. Motohashi established a general theorem which, roughly speaking, says that if (1) holds for two arithmetic functions then it also holds for their Dirichlet convolution; for precise assumptions and statement see [11]. So far, all methods depend on the large sieve inequality (see [10])

which sets the limit x1/2 for the modulus q in (1) and in its generalizations.

Let U0 = [(0, 1); and U1 = (0,1)]. Suppose we have a distribution of N points in , where, for k ≥ 1, is the unit cube consisting of the points y = (y1, … , yk+1) with 0 ≤ yi < 1 (i = 1 , … , k + 1). For X = (x1 ,…, xk + 1) in , let B(x) denote the box consisting of all y such that 0 ≤ yi < xi (i = 1 ,…, k + 1), and let denote the number of points of which lie in B(x).

A positive integer n is called a square-full integer, if p2 divides n, whenever p is a prime divisor of n. It is clear that each square-full integer can be written in the form a2b3, where a and b are positive integers; moreover, this representation is unique if we stipulate that b is square-free.

In an earlier paper [11] we have discussed the diophantine equation

and the more general equation

when l is an odd prime > 3, c a natural number (or more generally a rational number) satisfying several conditions, n also a natural number and x, y, z, non-zero rational integers. Other investigators have obtained quite a number of results concerning particularly the case n = 1. Obviously, without an essential restriction, we may assume that the natural number c contains no divisor a1n (since such a divisor can be absorbed in the power of z) and furthermore that x, y, z in (1) are relatively prime in pairs.

The result of my paper with the title given above (Mathematika, 26 (1979), 72–75) is not new; it was proved by Delone, see [2] and [3]. I have not been able to refer to either of these papers, but the result is given in [1], A similar problem was considered by Kitaoka in [5]. Professor Kneser of Göttingen tells me that he solved the problem about ten years ago in an unpublished manuscript; also that Schering's result [4] is less general and weaker than mine. I am obliged to Professor Peters of Münster for a copy of [1], giving the references [2] to [4], and also for the reference [5].

Consider a uniform body whose behaviour is described by the linear field equations of quasi-static, coupled thermoelasticity†:

This paper concerns conditions under which the classifying space functor transforms a fibre square of topological monoids into a fibre square of spaces. Before stating our results precisely, we first develop some relevant points concerning the continuity of functors.

Problems about partitions of cartesian products are common in mathematics. For example, in finite and infinite combinatorics, they keep emerging in Ramsay theory, where one seeks to show that, if a product is partitioned into finitely many parts, one part at least must contain a subset of a certain specified kind. In the transition from finite to infinite products, one usually imposes restrictions of a topological nature on the partition, in order to obtain theorems analogous to those which are valid in the finite case. (See, e.g., [G-P], [Si], [E].)

The concept of uniform regularity of a measure on a compact space—a property which allows uniform approximation of the measure on all compact sets—was introduced and discussed in [1], [2] and [3]. Further some extensions of the notion of uniform regularity are given in [4] and [6].

In recent work [2] we have investigated Borel isomorphisms at the first level, i.e. mappings that together with their inverses map Fσ-sets to Fσ-sets. We are grateful to Dr. F. Topsøe for writing to point out errors in the proofs of our Theorems 2.1, 2.2 and 2.3. The errors escaped our attention because we only wrote out the first step of a complicated inductive argument and carelessly and falsely claimed that the general step of the induction was similar to the first. To be more specific, in the proof of Theorem 2.1, although we do have

we do not, in general, have

Fortunately the proofs can be corrected.

In my recent paper [1], the corollary on p. 241 is not correct. It should be

Corollary. There is a sequence S, tending to infinity, such that for each t in S we have where γ is any zero of

Let FN be the set of real numbers x whose continued fraction expansion x = [a0; a1, a2,…, an,…] contains only elements ai = 1,2,…, N. Here N ≥ 2. Considerable effort, [1,3], has centred on metrical properties of FN and certain measures carried by FN. A classification of sets of measure zero, as venerable as the dimensional theory, depends on Fourier-Stieltjes transforms: a closed set E of real numbers is called an M0-set, if E carries a probability measure λ whose transform vanishes at infinity. Aside from the Riemann-Lebesgue Lemma, no purely metrical property of E can ensure that E is an M0-set. For the sets FN, however, metrical properties can be used to construct the measure λ.

Consider a homogeneous linear viscoelastic filament, one of whose ends is held fixed, while the other is subjected to a given traction τ(t) (t ≤ 0). I shall prove first that the total energy E(t), which is the sum of the kinetic and free energies, its derivative E′(t), and the rate of working W(t) satisfy

and

The quasigeostrophic equations describe large scale motion in the atmosphere and oceans at middle latitudes. Being considerably simpler than the primitive equations, they have been widely used for modelling atmospheric and oceanic circulation, and for studies of stability, frontogenesis and turbulence. A number of assertions have been made about these equations; first, that finite element approximate solutions converge in an open flow region [7]; secondly, that solutions depend discontinuously or nondeterministically on initial data [13]; and thirdly, that the energy wave number spectra of the solutions asymptote to the statistical equilibrium spectra of the spectrally truncated equations [12].

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grössencharakter Xi in Ki and consider the L-function

associated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined by

where

and a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.

It is the purpose of this note to draw attention to some connections between the topics mentioned in the title. These allow familiar results from one area of mathematics to be usefully exploited in another, to the mutual benefit of both. Moreover, the basic ideas suggest possible generalizations whose examination should prove worthwhile. In this introduction we give a brief account of the results to be derived and discussed in more detail in the later sections.