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.