It has been known since Vinogradov that, for irrational $\unicode[STIX]{x1D6FC}$, the sequence of fractional parts $\{\unicode[STIX]{x1D6FC}p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.

The object of this paper is to prove theorems of the following type.

THEOREM 1. If p and q are restricted to odd primes, anddenotes the Legendre symbol, then for x > 2

Among several interesting analogues of Chebyshev's problem about the largest prime factor of

there is the question of the largest prime factor of

where a is a given non-zero integer and the product is taken over positive primes p. The latter subject appears to have been first treated by Goldfeld [2] and Motohashi [6], who showed that, if Px be the greatest prime factor in question, then there exists a constant such that Px > xθ for all sufficiently large x. Their method, which involved the use of both Bombieri's theorem and the Brun-Titchmarsh theorem, had some affinity with the earlier treatments of Chebyshev's original problem.

Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

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 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.

Denote by N(t) the number of zeros β + iγ of ζ(s), s = σ + it, with 0 < β < 1 and 0 < γ < t. It is well known that

where

If(X1, …, Xn), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x1, …, xn) = 1, for suitable algebraic integers x1, …, xn. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X1, …, Xn) is any polynomial with relatively prime algebraic integers as coefficients, then f(x1, …, xn) is a unit, for suitable algebraic integers x1, …, xn. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x1, …, xn generate, as Watson could.

Let P be a (convex) d-polytope in the Euclidean space Ed and p a point of Ed not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of the boundary of P which is “visible” from p, i.e. the union of those facets whose supporting hyperplanes separate p and P, form a (d – l)-ball, whose boundary is a (d – 2)-sphere S (all balls, spheres and manifolds to be considered are piecewise-linear). The boundary-complex ℬ(P) induces a subdivision of S, which we call a (sharp) shadow-boundary ofP. is combinatorially isomorphic to the boundary complex of a poly tope which is a central projection of P from the point p on a hyperplane.

We consider the second order linear differential equation

where p and q are real-valued and p(t) > 0 for all t ≥ T. Our interest here is the oscillatory nature of solutions of (1.1). More particularly we consider the following questions, (I), (II) and (III).

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p}. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

Montgomery and Vaughan [12] have shown that the exceptional set in Goldbach's problem

satisfies

for some Δ>0. Li [10,11] has shown that we may take Δ = 0·079 and Δ = 0·086. If the Riemann Hypothesis is true for all Dirichlet L-functions then (1) holds for any Δ<½. This is a classical result due to Hardy and Littlewood [7].

If C and Co are two convex bodies in Ed we say that C slides (rolls) freely inside Co if the following condition is satisfied: for each x ∈ ∂C0 (and each rotation R) there is a translation t such that, if gC = C + t (= RC + t), then gC ⊂ Co and x ∈ ∂gC. This work establishes certain topological conditions which ensure the free rolling and sliding of C inside Co. One consequence of these conditions is that, if ∂K ∩ int gK is a topological ball for all rigid motions g, then K is a ball in the geometrical sense.

The asymptotic behaviour of a sequence of polynomials cm = cm(v) satisfying

is established. These polynomials occur in Hawkins' formula for the residues of a Bessel-zeta function at its possible poles in the left half plane. The results imply that cm(v)/cm(0) converges uniformly to cos πV on compact sets. This in turn implies that, for v not a half odd integer, all but finitely many of the possible poles are actual poles.

The properties of the solution of the differential equation governing the evolution of localised line-centred disturbances to a marginally unstable plane parallel flow were described by Hocking and Stewartson (1972). A corresponding study of the properties when the initial disturbance is point-centred is presented here. A localised burst at a finite time can be produced, for certain values of the coefficients which can be determined analytically. When the equation permits solutions with circular symmetry, two kinds of bursting solutions, as well as solutions which remain finite, are possible, but the boundary between bursting and finite solutions could not be determined analytically.

Several recent papers † have been devoted to the problem of the solubility in integers of a homogeneous equation (or of simultaneous equations), or the representation of an integer by a form (homogeneous polynomial). Such problems can be regarded as extensions of Waring's Problem: that of representing an integer as x1k+…+x8k with positive integral x1 …, x8. The methods used are developments of the Hardy-Littlewood method, or if not they employ auxiliary results proved by that method. The number of variables needed to ensure success is usually very large when the degree k of the homogeneous form is large. In this note we draw attention to a somewhat special problem for which a comparatively small number of variables, namely 2k+1, suffices.

The set of integers represented as the sum of three cubes of natural numbers is widely expected to have positive density (see Hooley [7] for a discussion of this topic). Over the past six decades or so, the pursuit of an acceptable approximation to the latter statement has spawned much of the progress achieved in the theory of the Hardy-Littlewood method, so far as its application to Waring's problem for smaller exponents is concerned. Write R(N) for the number of positive integers not exceeding N which are the sum of three cubes of natural numbers.

We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on $\mathbb{A}^{1}$ whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.

For a 3-connected graph G imbedded in the plane let νk(G) denote the number of k-valent vertices, and let pk(G) denote the number of k-gonal faces (countries) of the map defined by G. (We consider only graphs without loops or multiple edges; hence, νk(G) and pk(G) are defined only for k ≥ 3.) It is a wellknown consequence of Euler's formula that