Let N be a positive integer. We are concerned with the sum

Thus GN(N) is the ordinary Gauss sum. Previous methods of estimating such exponential sums have not brought to light the peculiar behaviour of GN(m) for m < N/2, namely that, for almost all values of m, GN(m) is in the vicinity of the point . A sharp estimate is given for max |GN(m)|, depending on the residue of N modulo 4. The results were suggested by graphs of GN(m) made for N near 1000. The analysis employs the Fresnel integrals and the Cornu spiral whose curvature is proportional to its arc length.

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.

Several years ago, Davenport [2] noticed that for any real irrational number ξ and for any prime number p, one at least of the numbers

has infinitely many partial quotients larger than p – 2 in its continued fraction expansion.

The object of the present note is to study functions f(z) which are bounded and analytic in |z| > 1, with Taylor expansion

where the sequence {λn} of integers satisfies the Hadamard gap condition

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.

Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.

Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let

be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.

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.

It is well known that two Borel subsets of the unit interval are Borel isomorphic, if, and only if, they have the same cardinality. The problem of the existence of analytic, non-Borelian subsets of the unit interval, which are not Borel isomorphic, has not been resolved within ZFC. With the additional assumption of the existence of an uncountable coanalytic set which does not contain a perfect set, it has been shown that there are at least three Borel isomorphism classes of analytic non-Borelian sets [4, 5].

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

§1. Let f(x) = x′Ax be a positive definite or semi–definite n-ary quadratic form with real symmetric matrix A. Then, f is Minkowski-reduced, if for all sets of integers m1, …, mn with gcd (mi, …, mn) = 1,

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.

If an integer does not have a k-th power of a positive integer, other than 1, for a divisor, it is said to be k–free. Let f(n) be an irreducible polynomial, with rational integer coefficients, of degree g, having no fixed k-th power divisors other than 1. We define

i.e. Nk(x) is the number of positive integers n not exceeding x such that f(n) is k-free. One would expect that f(n) is square-free for infinitely many n and further that, given x sufficiently large, there is an n with x < n ≤ x + h, such that f(n) is square-free for h = 0(x2) where ε is any real number > 0. These conjectures, however, seem to be extraordinarily difficult to prove. We begin with a brief account of the best results that have been attained so far.

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

Let ζ be a complex number. For n = 1, 2 … we write ωn (ζ for the supremum of the numbers ω for which

for infinitely many polynomials P of degree ≦ n with integer coefficients (H (P) denotes the maximum of the absolute values of the coefficients of P).

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.

In a compact group G, a sequence (Fn) of finite sets is uniformly distributed if the averaging operators

are uniformly convergent to the mean for continuous complex-valued functions f. In any compact metric group, there are uniformly distributed sequences of finite sets which are determined by a metric for the group. In some compact groups, there are uniformly distributed sequences of finite sets which are determined by the algebraic structure. A necessary and sufficient condition for a sequence of finite sets to be uniformly distributed in a compact metric group is that for any metric d for G and each εG, there is a sequence of one-to-one maps pn: Fn→ Fn such that

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.

In Hausdorff topological spaces there are currently three definitions of analytic sets due respectively to Choquet [1], Sion [8], and Frolīk [3, 4]. Here it is shown that these definitions are equivalent.

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.