In [4] we have given a simple method of estimating trigonometrical sums over prime numbers. Here we show how the argument can be adapted in order to give estimates for the distribution of αp modulo 1 which are sharper than those obtained by I. M. Vinogradov [5], [6]. Vinogradov uses the sieve of Eratosthenes to relate the sum

to the bilinear form

the function μ being the Mobius function. When d1 … ds is small compared with N this can be treated in a fairly straightforward manner. However, in order to treat the terms with d1 …ds close to N, Vinogradov has to introduce an argument of a rather recondite combinatorial nature.

I discuss various necessary and sufficient conditions for a K-analytic space to be Souslin. In particular, I show that if the continuum hypothesis is true, then there is a non-Souslin K-analytic space in which every compact set is metrizable; while if Martin's Axiom is true and the continuum hypothesis is false, this is impossible.

We let K(a1a2, ak) denote the continuant formed from the positive integers a1 …, ak; that is,

Of course, K(a1, …, an) is the denominator of the continued fraction

We use the convention that K (empty set) = 1.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.

Denote by ζk′(S), the derivative of the Dedekind zeta-function associated with the real quadratic field K. Then it is known that

where ζ(S) is the Riemann zeta-function, and L(s, x) is the Dirichlet L-series associated with the Legendre symbol X. Moreover, we have the functional equation

where D is the discriminant of K.

The Fourier transform f of a function is defind by

t ∊ ℝn. A(ℝn) is the isometric image of L1(ℝn) under the Fourier transformation. We extend the Fourier transformation to a mapping of ℒ(ℝn) to itself and denote by A′(ℝn) the isometric image of L∞(ℝn).

An asymptotic expansion is obtained for this sequence, of interest in combinatorial analysis. Values are given for the constants appearing in the leading term and a numerical comparison made.

Littlewood [5, Problem 4.19, originally 4] conjectured that there is an absolute constant C > 0 such that, for every sequence of distinct integers n1, n2, n3, …, if

then

Cohen [2] showed

for some absolute constant C, with b = 1/8. Davenport [3] gave a more explicit version of Cohen's proof and improved the estimate to b = 1/4. Pichorides [6] added another refinement to obtain b = ½, and has, more recently, obtained ‖fN‖1>C(log N)1/2. This seems to be the best estimate so far without restriction on the sequences. We shall show that the methods of Davenport and Pichorides may be extended to obtain better results for certain classes of sequences. Specifically, we prove the following theorems.

Let denote the class of all compact convex sets in Euclidean n-dimensional space En, and let y be the collection of those members of k which are centrally symmetric. The topology in is that induced by the Hausdorff metric.