We shall give a simple new proof of the following known theorem [1, 2].

Theorem. The upper density of a packing of translates of a convex disc cannot exceed the density of the densest lattice-packing of these discs.

In [] and [2] this theorem is proved for centrally symmetric discs. The general case can be reduced to this one by applying to the discs the known construction of central symmetrization. Our proof goes in a reverse way. We shall give a direct proof for a special family of asymmetric discs whose centrally symmetric images exhaust the family of centrally symmetric convex discs. Using the properties of the symmetrization, this implies the validity of the theorem for all centrally symmetric discs, and consequently for all convex discs. This procedure of going from a special case to the general one, by applying the symmetrization twice, is illustrated by the following example. The validity of the theorem for a Reuleaux triangle implies its validity for a circle, which implies its validity for any disc of constant width.

Let η(z) be the Dedekind eta function defined by

where q = exp(2πiz) and Im (z) > 0. Then it is famous that

One of the oldest questions concerning primitive roots is that of the actual order df magnitude of the least positive primitive root m(p) to a large prime modulus p. During the last sixty years the interest of several mathematicians has been attracted by this problem. The history of the major results is too well known to need elaboration. The best known bound for m(p) is due to Wang [8] and Burgess [1] who independently obtained

As usual, if π is a finite group and ℳ ⊆ ℚπ a maximal order containing , we set

This is well known to be a finite group, and independent of the choice of ℳ.

Let F be a number field, l a prime, and K a normal extension of F for which Gal(K/F) is topologically isomorphic to the additive group Zl. Then, corresponding to the subgroups ln Zl, there is a chain of fields F = F0 ⊂ F1 ⊂ F2 ⊂ … ⊂ K so that Gal (Fn/F) is cyclic of order ln. Let An be the l-Sylow subgroup of the ideal class group of Fn. By the now well-known theorem of Iwasawa, there are constants μ, λ and ν so that, for all sufficiently large n, |An| = len, where en = μln + λn + ν. Much work has been done investigating these constants, especially showing μ to be 0 for cyclotomic Zl-extensions of abelian fields F, [2], and computing λ and ν in special cases. In many cases, it has been shown that μ = λ = 0, and I know of no cases where μ or λ are positive when F is totally real (cf. Greenberg [3]). Here we investigate the maps in,m: An → Am induced by the inclusion of Fn in Fm for m ≥ n, with a view to determining their kernels. When μ = λ = 0, we prove that in, n+s(An) = for large enough n, and that, for large enough n, Ker in, n+s coincides with the kernel of the ls power map on An. I thank A. Brumer for helpful conversations on this matter.

Let be a smooth projective, geometrically irreducible curve over a finite field . We fix a rational point ∞on , and consider the ring A of functions on regular away from ∞. We set k to be the function field of and k∞ its completion at ∞. After taking algebraic closure we obtain the field whose elements will be called “numbers”.

A prime p > 2 is called irregular, if it divides the numerator of at least one of the Bernoulli numbers B2, B4, …, Bp – 3 (in the even suffix notation). The study of irregular primes has its origin in the famous theorem of Kummer which states that p divides the class number of the p-th cyclotomic field, if, and only if, p is irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite.

In line with the Ritt–Seidenberg elimination theorem in differential algebra [RIT], [SEI], and with an “approximation theorem” by Denef and Lipshitz [DEL] for formal power series, and with an elimination theorem by the author [RUB1] for C∞ solutions of systems of algebraic differential equations (ADE's), one is led to consider the corresponding elimination question for Cn solutions. Somewhat in the spirit of [RUB2], though, we produce a negative result in this direction.

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 ( = 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

A linear second order partial differential equation with variable coefficients is considered. The equation is relevant in a number of physical situations. Simple general solutions are obtained subject to the coefficients satisfying certain constraints.

It was shown by Vinogradov (see Theorem 7, Chapter 4 of [18]) that, for ε > 0, there are infinitely many solutions in primes p of the inequality

where {x} denotes the fractional part of x and y = 0.1. The value of γ was improved to

by Kaufman [9]. On the Riemann Hypothesis he showed that one can take γ =¼ The method used actually shows that, for any real β and any δ with 0 < 8 > 1, the number of primes p ≤ x satisfying

is

where π(x) denotes the number of primes not exceeding x. The sequence √p is, of course, a subsequence of the sequence n½ whose distribution modulo one has also been investigated (see Chapter 2, Section 3 of [10]: the argument for sequences nσ (0 < σ < 1) is entirely elementary). It is useful in this context to define the discrepancy (modulo one) of a sequence an by

§1. Let E = E(A) be the set of real numbers x ε (0, 1) whose regular continued fraction expansion

contains only partial denominators ai from a given set A of positive integers. For finite A the (Hausdorff-) dimensional numbers dim E have been studied by I. J. Good ‘2’ and T. W. Cusick ‘1’. C. A. Rogers ‘8’ introduced a natural probability measure on E. He showed that excluding sets with measure between 0 and 1 (in the strict sense) from E does not reduce the dimensionality more than excluding sets of measure zero, and that the minimal (“essential”) dimensions ess dim E arising in this way is smaller than dim E, at least for A = {1,…, r} when r = 2 or r is sufficiently large.

Let ℱ be a set of m subsets of X = {1,2,…, n}. We study the maximum number λ of containments Y ⊂ Z with Y, Z ∊ ℱ. Theorem 9. , if, and only if, ml/n → 1. When n is large and members of ℱ have cardinality k or k–1 we determine λ. For this we bound (ΔN)/N where ΔN is the shadow of Kruskal's k-cascade for the integer N. Roughly, if m ∼ N + ΔN, then λ ∼ kN with infinitely many cases of equality. A by-product is Theorem 7 of LYM posets.

The purpose of this paper is to prove the inequality of Theorem 1. The problem is due to B. Korenblum, who asked about it in connection with the characterization of the zero sets of functions analytic in the unit disc satisfying a growth condition. The problem was communicated to us by W. K. Hayman.