The n-th roots of unity 1, ω, …, ωn-1, where ω = exp (2πi\n), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the linear relations between them with integral coefficients. Let U denote the vector space of elements u = (u0, …, un−1) over Q and let N be the subspace of elements u defined by the relation u0+u1ω+…+un−1ωn−1=0. (1)

The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations

with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing

in which e(α) denotes e2πiα, we observe that

where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

We prove algebraic transformations for the generating series of three Apéry-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method that derives three new series for 1/π from a classical Ramanujan’s series.

Let k denote a fixed natural number with k > 2, let ℛs(n) denote the number of representations of n as the sum of sk-th powers of natural numbers, and let

with

denote the corresponding singular series.

Let be a = ℤ-order in A, a finite dimensional Q-algebra. K0() denotes the Grothendieck group of projective right -modules. locally isomorphic to } is a subgroup of K0() and is called the locally free classgroup of . (If = ℤΓ for some finite group Γ then as all ℤΓ projectives are locally free [12].)

Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K = r (mod 2 π), on [-1, 1 ] and on ℝ. The key result is the establishment of the inequality

for every rational function f=pn/qn, where pn is a polynomial of degree at most n with complex coefficients and

with | aj | ≠ 1 for each j and for every zo∈ δ D, where δ D,= {z∈ ℂ: |z| = l}. The above inequality is sharp at every z0∈δD.

Let $\unicode[STIX]{x1D713}:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ be a non-increasing function. A real number $x$ is said to be $\unicode[STIX]{x1D713}$-Dirichlet improvable if it admits an improvement to Dirichlet’s theorem in the following sense: the system

$$\begin{eqnarray}|qx-p|<\unicode[STIX]{x1D713}(t)\quad \text{and}\quad |q|<t\end{eqnarray}$$

$t$

$D(\unicode[STIX]{x1D713})$

$D(\unicode[STIX]{x1D713})^{c}$

$\unicode[STIX]{x1D713}$

The Minkowski-Hlawka bound implies that there exist lattice packings of n-dimensional “superballs” |x1|σ + … + |xn|σ ≤ 1 (σ = 1,2,…) having density Δ satisfying log2 Δ ≥ −n(l + o(l)) as n → ∞. For each n = pσ (p an odd prime) we exhibit a finite set of lattices, constructed from codes over GF(p), that contain packings of superballs having log2 Δ ≥ −cn(l + o(l)), where for σ = 2 (the classical sphere packing problem), worse than but surprisingly close to the Minkowski-Hlawka bound, and c = 0·8226 … for σ = 3, c = 0·6742 … for σ = 4, etc., improving on that bound.

The object of this paper is first to generalize the basic inequality of the large sieve method to exponential sums in many variables, and then to deduce results for algebraic number fields that are analogous to known results for the rational field.

We are interested in the distribution of those zeros of the Riemann zeta-function which lie on the critical line ℜs = ½, and the maxima of the function between successive zeros. Our results are to be independent of any unproved hypothesis. Put

Let λ(n) be Liouville's function denned by

where v is the number of prime factors of n, repeated factors being counted according to their multiplicity. Alternatively, λ(n) may be denned by the relation

where ζ(s) is the zeta function of Riemann.

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hörmander-Rådström lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝn is contained in another convex body. This generalizes a theorem of Weil (cf. [10]).

Let R be a real closed field and a variety of real dimension k′ which is the zero set of a polynomial Q∈R[X1,…, Xk] of degree at most d. Given a family of s polynomials = {P1,…, Ps}⊂R[X1,…,Xk] where each polynomial in has degree at most d, we prove that the number of cells defined by over is (O(d))k Note that the combinatorial part of the bound depends on the dimension of the variety rather than on the dimension of the ambient space.

We shall establish here a new relation between the classnumber h of an algebraic number field K and the signature of its group of units Y.

Write dim2(A) for the number of even invariants of a finite Abelian group A. Denote by C the absolute classgroup of K (ideals modulo principal ideals), and by P the quotient group of principal ideals modulo totally positive principal ideals.

Let R be a Dedekind domain whose quotient field K is an algebraic number field, and let Λ be an R-order in a semisimple K-algebra A with 1. A Λ-lattice is a finitely generated R-torsionfree left Λ-module. We shall call a Λ-lattice M locally free of rank n if for each maximal ideal p of R, Mp is Λp,-free on n generators. (The subscript p denotes localization.) The (locally free) class group of Λ is the additive group C(Λ) generated by symbols

where

and where xM = 0 if and only if M is stably free (that is, M + Λ(k) ≅ Λ + Λ(k) for some k).

A number of known estimates of the number of translates, or lattice translates, of a convex body H required to cover a convex body K are obtained as consequences of two simple results.

Rather more than thirty years ago Erdős and Mirsky [2] asked whether there exist infinitely many integers n for which d(n) = d(n + 1). At one time it seemed that this might be as hard to resolve as the twin prime problem, see Vaughan [6] and Halberstam and Richert [3, pp. 268, 338]. The reasoning was roughly as follows. A natural way to arrange that d(n) = d(n + l) is to take n = 2p, where 2p + 1 = 3q, with p, q primes. However sieve methods yield only 2p + 1 = 3P2 (by the method of Chen [1]). To specify that P2 should be a prime q entails resolving the “parity problem” of sieve theory. Doing this would equally allow one to replace P2 by a prime in Chen's p + 2 = P2 result.

Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular invariant j(w); he calls j(w) a class invariant. He goes on in §§125-144 to consider the values f(w) of other modular functions f(z); he shows that in certain cases the degree of the extension ℚ(ω, f(ω)) of ℚ(ω, j(ω)) is much less than that of ℚ(f(z)) over ℚ(f(z)); indeed, f(ω) is often in ℚ(ω, j(ω)), and in such circumstances Weber calls f(ω) a class invariant too. Using such results, Weber computes many class invariants—an end in itself, since the numbers are so beautiful. More recently, results of this type have been applied to determine all the complex quadratic fields with class number 1, and to prove that elliptic curves of certain families always have infinitely many rational points-see [9, 2, 5].

The stability against small disturbances which do not bend the vortex lines of a circular cylindrical vortex sheet of variable radius is examined. If the jump in tangential velocity across the sheet falls off with time faster than (time)–íthe disturbances are bounded (or grow less rapidly than linearly with time) so that the sheet is effectively stable. If the tangential velocity jump falls off with time slower than (time)–í the sheet is unstable.

We determine what is the maximum possible (by volume) portion of the three-dimensional Euclidean space that can be occupied by a family of non-overlapping congruent circular cylinders of infinite length in both directions. We show that the ratio of that portion to the whole of the space cannot exceed π/√12 and it attains π/√12 when all cylinders are parallel to each other and each of them touches six others. In the terminology of the theory of packings and coverings, we prove that the space packing density of the cylinder equals π/√12, the same as the plane packing density of the circular disk.