In [4] we have given a simple method of estimating trigonometrical sums over prime numbers. Here we show how the argument can be adapted in order to give estimates for the distribution of αp modulo 1 which are sharper than those obtained by I. M. Vinogradov [5], [6]. Vinogradov uses the sieve of Eratosthenes to relate the sum

to the bilinear form

the function μ being the Mobius function. When d1 … ds is small compared with N this can be treated in a fairly straightforward manner. However, in order to treat the terms with d1 …ds close to N, Vinogradov has to introduce an argument of a rather recondite combinatorial nature.

§1. Let pi denote the i-th prime number, and let

The prime number theorem implies Er ≤ r. Bombieri and Davenport [1] showed that

improving earlier results of Erdős, Rankin and Ricci. Their basic result (corresponding to Lemma 1 below) counts pairs of primes differing by 2n with some weight t(n).

An arithmetic function f(n) is said to be additive, if it satisfies the relation f(ab) = f(a) + f(b), for every pair of coprime integers a and b; and stronglya dditive if, in addition, f(pm) = f(p) for every prime-power pm.

We let K(a1a2, ak) denote the continuant formed from the positive integers a1 …, ak; that is,

Of course, K(a1, …, an) is the denominator of the continued fraction

We use the convention that K (empty set) = 1.

In this paper we continue our study of the values taken by Euler's ø-function begun in [1]–[3]. Let ør(n) be the iterated ø-function, that is ør(n) = ø{ør-1(n} where ø1 = ø1. Let

The question, whether a given element of a C*-algebra has an image of rank one in some faithful representation, was studied in [3]. Such elements were characterised there by the property of being “single” (as defined below). As was pointed out in [3], Section 5, this criterion fails for general Banach algebras and the purpose of this paper is to provide a stronger condition giving the required representation property for any semi-simple Banach algebra.

Littlewood [5, Problem 4.19, originally 4] conjectured that there is an absolute constant C > 0 such that, for every sequence of distinct integers n1, n2, n3, …, if

then

Cohen [2] showed

for some absolute constant C, with b = 1/8. Davenport [3] gave a more explicit version of Cohen's proof and improved the estimate to b = 1/4. Pichorides [6] added another refinement to obtain b = ½, and has, more recently, obtained ‖fN‖1>C(log N)1/2. This seems to be the best estimate so far without restriction on the sequences. We shall show that the methods of Davenport and Pichorides may be extended to obtain better results for certain classes of sequences. Specifically, we prove the following theorems.

The Fourier transform f of a function is defind by

t ∊ ℝn. A(ℝn) is the isometric image of L1(ℝn) under the Fourier transformation. We extend the Fourier transformation to a mapping of ℒ(ℝn) to itself and denote by A′(ℝn) the isometric image of L∞(ℝn).

Let denote the class of all compact convex sets in Euclidean n-dimensional space En, and let y be the collection of those members of k which are centrally symmetric. The topology in is that induced by the Hausdorff metric.

An incident sound field is scattered by a semi-infinite rigid screen with periodically arranged slits or circular apertures and an approximate solution is sought when the slit (or aperture) width is small and the wavelength is large compared with the separation. An integral equation formulation is used to show that the scattering properties of the screens are equivalent to those of a homogeneous compliant plate. The effective compliance is estimated and is found to be essentially uniform over the plate, with corrections close to the edges.

Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.

Denote by ζk′(S), the derivative of the Dedekind zeta-function associated with the real quadratic field K. Then it is known that

where ζ(S) is the Riemann zeta-function, and L(s, x) is the Dirichlet L-series associated with the Legendre symbol X. Moreover, we have the functional equation

where D is the discriminant of K.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

I discuss various necessary and sufficient conditions for a K-analytic space to be Souslin. In particular, I show that if the continuum hypothesis is true, then there is a non-Souslin K-analytic space in which every compact set is metrizable; while if Martin's Axiom is true and the continuum hypothesis is false, this is impossible.

An asymptotic expansion is obtained for this sequence, of interest in combinatorial analysis. Values are given for the constants appearing in the leading term and a numerical comparison made.

Let Xt be a Lévy process in Rd, d dimensional euclidean space. That is Xt is a strong Markov process whose transition function, P(t, x, A), satisfies: