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

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

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.

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 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.

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

be a cusp form of even integral weight k > 2 for the full modular group. Then the Dirichlet series

is absolutely convergent for σ > ½(k + 1). Hecke showed that LF is an entire function of s satisfying the functional equation

Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.

Let q = pn, p a rational prime, and let be the finite field with q elements. The polynomial ring is considered as an analogue of the ring of rational integers ℤ. Completing the quotient field with respect to the normalized valuation at ∞, and then taking algebraic closure, we obtained the field k∞ whose elements will be called “numbers”.

We consider the Falkner–Skan equation

in the special case β = – 1. It is known that this equation may be integrated twice to get the Riccati equation

Before turning to the questions to be considered in this paper, we recall two other problems. Let C(a, p) be the class of all convex discs of area not less than a given constant a and perimeter not greater than a given constant p. What is the densest packing and what is the most economical covering of the Euclidean plane with discs from C(a, p)?

Both problems are interesting only if p2/a < 8√3, i.e. if p is less than the perimeter of a regular hexagon of area a. In this case, the densest packing arises from a regular hexagonal tiling by rounding off the corners of the tiles by equal circular arcs so as to obtain smooth hexagons of area a and perimeter p.

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.

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

The classical mean value theorem for Dirichlet's polynomials states that

see H. L. Montgomery [7]. This formula is very useful in the theory of the Riemann zeta-function ζ(s). From the approximate functional equation

where | χ(½ + it)| = 1, u, v ≥ 1, 2πuv = t (see E. C. Titchmarsh [8]) it follows that χ(½ + it) can be well approximated by Dirichlet's polynomials of length N< t½.

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

Let be k non-overlapping translates of the unit d-ball Bd in euclidean d-space Ed. Let Ck denote the convex hull of their centres and let Sk be a segment of length 2(k– 1). Furthermore, let Vd denote the d-volume. L. Fejes Tóth conjectured in [1], that, for d ≥ 5,

Let f be a non-zero cusp form of weight k on SL(2, ℤ) with Fourier expansion We assume further that f is normalized (a(1) = 1) and that f is an eigenfunction of the Hecke operators. Define