Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made to obtain tight estimates for the number of lattice points of a given lattice in convex bodies in terms of the basic quermass-integrals Wo,…, Wd, whose eminent role shows in Hadwiger's functional theorem [14, 15, 16, see also 17, p. 221–225]. (For the discrete analogues of Wo,…, Wd see [2].) The present paper is concerned with an upper estimate of this kind.

The Picone identity which occurs in the oscillation theory of second-order ordinary differential equations is

We prove that the relation type of all high powers of an ideal in a Noetherian ring is either one or two. It is one exactly when some power of the ideal is locally principal.

We shall consider incomplete exponential sums of the shape

where q, a and h are integers satisfying 1 ≤ a < a + h ≤ q, f(x) is a function denned at least for the integers in the range of summation, and eq(t) is an abbreviation for e2πit/q.

We consider the size of large character sums, proving new lower bounds for Δ(N,q)=sup χ≠χ0 mod q∣∑ n<Nχ(n)∣ in almost all ranges of N. The proofs use the resonance method and saddle point analysis.

Dyadic shifts $D\oplus T$ of point distributions $D$ in the $d$-dimensional unit cube $U^{d}$ are considered as a form of randomization. Explicit formulas for the $L_{q}$-discrepancies of such randomized distributions are given in the paper in terms of Rademacher functions. Relying on the statistical independence of Rademacher functions, Khinchin’s inequalities, and other related results, we obtain very sharp upper and lower bounds for the mean $L_{q}$-discrepancies, $0<q\leqslant \infty$. The upper bounds imply directly a generalization of the well-known Chen theorem on mean discrepancies with respect to dyadic shifts (Theorem 2.1). From the lower bounds, it follows that for an arbitrary $N$-point distribution $D_{N}$ and any exponent $0<q\leqslant 1$, there exist dyadic shifts $D_{N}\oplus T$ such that the $L_{q}$-discrepancy ${\mathcal{L}}_{q}[D_{N}\oplus T]>c_{d,q}(\log N)^{(1/2)(d-1)}$ (Theorem 2.2). The lower bounds for the $L_{\infty }$-discrepancy are also considered in the paper. It is shown that for an arbitrary $N$-point distribution $D_{N}$, there exist dyadic shifts $D_{N}\oplus T$ such that ${\mathcal{L}}_{\infty }[D_{N}\oplus T]>c_{d}(\log N)^{(1/2)d}$ (Theorem 2.3).

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

Let $L(s,\unicode[STIX]{x1D712})$ be the Dirichlet $L$-function associated to a non-principal primitive character $\unicode[STIX]{x1D712}$ modulo $q$ with $3\leqslant q\leqslant 400\,000$. We prove a new explicit zero-free region for $L(s,\unicode[STIX]{x1D712})$: $L(s,\unicode[STIX]{x1D712})$ does not vanish in the region $\mathfrak{Re}\,s\geqslant 1-1/(R\log (q\max (1,|\mathfrak{Im}\,s|)))$ with $R=5.60$. This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$.

Let $E(N)$ denote the number of positive integers $n\leqslant N$, with $n\equiv 4\;(\text{mod}\;24)$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

This paper presents an analysis of two canonical problems for the steady supersonic inviscid flow past a wing-body combination with the body nonlifting and the wings at a very small angle of attack. Pressure distributions in the vicinity of the lines separating the interaction regions from the remainder of the wing and body are determined from asymptotic expansions by modifying a method due to Friedlander, who considered the diffraction of sound pulses by a cylinder. In a manner similar to Keller's geometrical theory of diffraction the results of the canonical problem are directly applicable to the case where the cross section of the body is any closed convex curve which meets the wings at right angles. In the first canonical problem, the leading edge of the wings has no sweepback, that is, it is normal to the body surface. Here results are compared with those of an approach using boundary layer methods by Stewartson and of a Green's function method by Jones. In the second problem which is even more important in practice and has been neglected in the literature, the leading edge of the wings has positive sweepback; our results are valid provided that the leading edge of the wings is supersonic.

Let λ1, …, λ8 be any non-zero real numbers not all of the same sign and not all in rational ratios. According to a theorem of Davenport and Roth [3], given a real number κ, the inequality

has infinitely many solutions in positive integers for any ε > 0. Recently Liu, Ng and Tsang [5] gave a refinement of this result: for any δ > 0, the inequality

has infinitely many solutions in positive integers. In the present note we obtain a better exponent.

In many physical problems it is necessary to express a solution of Laplace's equation relative to one set of coordinates in terms of harmonics relative to another set. We term such a relationship an addition or shift formula. Well-known examples are the formulae for spherical harmonics [Hobson 1, p. 139] and cylindrical harmonics [Watson 2, p. 360]. In this paper we shall consider the problem for oblate spheroidal coordinate systems and obtain some addition theorems for the corresponding harmonics. These addition theorems are used to write down “two-centred” expansions of the Coulomb Green's function, and by a limiting process a new form of the “one-centred” expansion is obtained in a non-orthogonal coordinate system, closely related to oblate spheroidal coordinates. This expansion is applied to the evaluation of the Coulomb (or gravitational) energy of spheroidal distributions of charge (or mass) in which the surfaces of constant density are concentric similar spheroids, a situation which occurs in both nuclear physics and cosmology [Carlson 3]

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

A study is made of the solutions of the differential equation f′″ + ff″ = 1 subject to the initial conditions f(0) = f′(0) = 0.

We obtain density estimates for subsets of the $n$-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid $\{1,2,\dots ,N\}^{2}$ lacking four corners in a square has size at most $\mathit{CN}^{2}(\log \log N)^{-c}$. Our proofs involve the density increment method of Roth [J. London Math. Soc.28 (1953), 104–109] and Gowers [Geom. Funct. Anal.11(3) (2001), 465–588], together with the $U^{3}$-inverse theorem of Green and Tao [Proc. Edinb. Math. Soc. (2) 51(1) (2008), 73–153].

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.

We show that if d≥4 is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

A line that intersects every member of a finite family F of convex sets in the plane is called a common transversal to F. In this paper we study some basic properties of T(k)-families: finite families of convex sets in the plane in which every subfamily of size at most k admits a common transversal. It is known that a T(k)-family admits a partial transversal of size α∣F∣ for some constant α(k) which is independent of F. Here it will be shown that (2/(k(k−1)))1/(k−2)≤α(k)≤((k−2)/(k−1)), which are the best bounds to date.

Given some unit vectors a1,…,am∈ℝn that span all ℝn and some positive numbers θ1,…,θm, we consider for every p≥1 the convex body We will give some upper bounds for the volume of Kp and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp–Lieb inequality. We will also see how the best choice of these parameters is related to the transformation which takes Kp to a special position which, for instance, when p=∞, is John’s position.