Let V(x) denote the number of distinct values not exceeding x taken by Euler's ø-function, so that we have π(x) ≤ V(x) ≤ x. It was shown by Erdős and Hall [1] that for each fixed B > 2√(2/log 2), the estimate

holds; moreover we stated that the ratio V(x)/π(x) tends to infinity with x, faster than any fixed power of log log x. Our aim in the present paper is to prove the following result.

One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + h], averaged over n ≤ N, tends to the limit λ, when N and h tend to infinity in such a way that h ∼ λ log N, with λ a positive constant.

Some sixty years ago Hardy and Ramanujan [6]introduced the notion of normal order of an arithmetic function.

A real-valued arithmetic function f)n) is said to have a normal order if there is a function g(n), which is non-negative and non-decreasing for all sufficiently large integers n, so that, for each fixed ε > 0, the integers n, for which the inequality

is satisfied, have asymptotic density zero. Thus, in this certain sense, f(n) behaves almost surely like g(n). We say that f(n) has the normal order g(n). In their original paper Hardy and Ramanujan asked that the function g(n) be “elementary”, but this is a requirement that subsequent researchers have dropped.

In this paper we are interested in two related measures of the degree of approximation of a complex number ζ by algebraic numbers. For a given integer n ≥ 1, write wn(ζ) for the supremum of the exponents w for which

for infinitely many polynomials

in Z[x] of height H(P) = max |av|. Clearly 0 ≤ w1 (ζ) ≤ w2 (ζ) ≤ …. On the other hand, write for the supremum of the exponents w for which

for infinitely many algebraic numbers α of degree at most n.

The object of this paper is to give a new characterization of the set of Pisot-Vijayaraghavan numbers (P.V.-numbers for short). As usual, [x] and [x] represent respectively the integer part and the fractional part of the real number x, and

The stability of a two-dimensional vortex sheet against small disturbances in the plane of flow is examined. An integro-differential equation for the disturbances is derived and the possibility of solving it approximately is discussed. The approximation is equivalent to saying that short waves grow in a fashion determined by the local strength of the vortex sheet and it is shown that this need not be true throughout the evolution of the disturbance unless the growth rate is the same everywhere. It is possible for the disturbance on a distant part of the vortex sheet to control what happens locally, if the disturbance on the distant part is growing more rapidly. The approximate theory is applied to the tightly-wound spirals of aerodynamic interest and these are shown to be stable.

Let T be a nonlinear Volterra integral operator of the form

(with I compact, a < b), whose kernel K = K(x, t, u) satisfies the following conditions:

with

In [1], P. X. Gallagher introduced a new sieve which is designed to produce better estimates than the large sieve when a vast number of congruence classes are chosen for each of the sieving primes. By making more explicit use of the principles underlying the sieve, these estimates can be improved and generalized to the case of complex quadratic fields. We shall see that the resulting estimates are best possible, if there are only a bounded number of unsieved classes per prime.

A length function, for a group, associates to an element x a real number |x| subject to certain axioms, including a cancellation axiom which embodies certain cancellation properties for elements of a free group. Integer valued length functions were introduced by Roger Lyndon [1] where, with a more restrictive set of axioms than ours, it is shown that a length function for a group is given by a restriction of the usual length function on some free product.

The two inter-related aspects of this laminar flow study are, first, the effects of indentations of length O(a) and height O(aK-⅓) on an otherwise fully developed pipeflow and, second, the manner in which such a pipeflow adjusts ahead of any nonsymmetric distortion to the downstream conditions. Here K is the typical Reynolds number, assumed large, and a is the pipewidth. The flow structure produced by the particular slowly varying indentation, or by a suitable distribution of injection, comprises an inviscid core, effectively undisplaced, and a viscous wall-layer, where the swirl velocity attains values much greater than in the core and where the nonlinear governing equations involve the unknown pressure force. Linearized solutions for finite-length, unbounded or point indentations, and for finite blowing sections (which model the influence of a tube-branching), demonstrate the upstream influence inherent in the nonlinear problem, for steady or unsteady disturbances. It is suggested that the upstream interaction caused there provides the means for the upstream response in the general case where the indentation, say, produces a finite constriction of the tubewidth.

The problem of finding an axiomatic characterisation of dimension was first tackled by Menger, who gave a set of five independent axioms characterising the dimension (in the sense of dim, ind, or Ind since they are all equal on separable metric spaces) of subsets of the plane [7, p. 156]. The question of whether Menger's axioms characterise the dimension of more general spaces is still unsettled. Recently, Nishiura “11” obtained a set of seven independent axioms characterising the dimension of separable metric spaces. By modifying one of Nishiura's axioms, Aarts [1] then obtained an axiomatic characterisation of the strong inductive dimension (Ind) of metric spaces. Also, Ščepin [12] and Lokucievskiĭ [9] have obtained different axioms for dim on the class of compact metric and compact spaces, respectively. We present here four sets of independent axioms that characterise the dimension function u-dim, which is defined on the class of all uniform spaces.

Let K be a field and * an involution of K. Let V and V′ be vector spaces over K of dimensions greater than or equal to 3, and let P(V) and P(F′) denote the projective spaces associated with V and V′ respectively. The fundamental theorem of projective geometry states that a bijection σ between P(V) and P(V′), which preserves the collinearity of points in P(V) and P(V′), is induced by a semi-linear bijection ø between V and V′ with respect to an automorphism α of K. We now consider the following additional structure on V and V′. Let f and f′ be hermitian forms with respect to * on V and V′ respectively; they define twisted polarities in P(V) and P(V′). We prove the following theorem. If there are no self-polar points in P(V) and P(V′) and σ preserves the twisted polarities, then ø is “almost” an isomorphism of the hermitian forms in the sense that

for some non-zero A in K such that A = A*. We give examples of forms f and f′ satisfying the above condition and we illustrate our theorem.

The 2-adic density of a quadratic form is a factor in the expression for the weight of a genus (of positive forms) and so has been investigated by a number of writers. As pointed out by Pall [1], the most complete investigation, that by Minkowski, omitted many details and errors resulted. Unfortunately, Pall's paper also contains errors: the last — in his formula (23) should be +, and 8, 4 in the second and third lines after (47) should be 3, 2. In the fourfold distinction of cases (with nine sub-cases) at the end, the possibility ei+l – ei = 1, øi and øi+1 both pp., seems to have been overlooked.

We give sufficient conditions for the spectra and essential spectra of certain classes of operators to be contained in or coincide with an interval of the form [μ, ∞).

Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.