This note sets out from the observation that it is not, in general, possible to express a homogeneous cubic polynomial in five variables as a sum of cubes of seven linear forms. Some of the geometry, to which particular cubics which do happen to be so expressible give rise, is described. Further particularisations are mentioned, and one such cubic investigated in some detail.

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

Here we give necessary and sufficient conditions foi a prime ι to divide the class number of the Galois closure of a pure field of degree ι over the rationals. The work extends that of Honda in [4] and that of the first author in [8].

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.

§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,

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.

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.

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 the above paper [1] there is an error in the statements of Proposition 1 and Theorem 1 which we shall now correct. The principal results, Theorems 2 and 3, are unaffected.

§0. Einführung. Durch die bekannten Croftonschen Integrale können bekanntlich die Minkowskischen Quermaβintegrale konvexer Körper dargestellt werden. In der vorliegenden Note betrachten wir gewisse Erweiterungen dieser klassischen integralgeometrischen Formeln, durch die allgemeinere invariante Eikörperfunktionale gegeben sind. Es handelt sich hierbei um kinematische Integrale mit beweglichen unterdimensionalen Teilräumen, wobei passend gewahlte Funktionen ihrer Abstände vom Eikörper eingehen.

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].

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.

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.

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

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 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.

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.

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.