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.

Novikov [7] or [5, Th. 2, p. 510] proves that: if{An} is a sequence of analytic sets in a complete separable metric space and

then there is a sequence {Bn} of Borel sets with

Let ℝ∞ be the direct limit of the Euclidean spaces ℝn. Now the orthogonal group O(∞) acts on ℝn and the direct limit O(∞) of the groups O(∞) acts on ℝ∞. The infinite pin group Pin(∞) is an extension of ℤ2 by O(∞) and admits the following presentation: the generators are the unit vectors xf in ℝ∞ and the relations are

Let K be a successor cardinal. Call an regularizing family (for short, a regularizing family) if Card ℱ = k and for each with Card . In [7], Prikry proved, assuming V = L, that if is an ultrafilter on K which is uniform (Card X = K, each ) then is regular (some forms a regularizing family).†

An internal wave motion, below a layer of uniform fluid, induces a weak current on the free surface in the form of a long wave with phase velocity cI. A uniform progressive train of surface waves, whose wave-length is much shorter than that of the current is incident on it from infinity and undergoes modification. In particular, when the group velocity cg of the progressive wave is equal to cI, the resonance takes place and then, even though the amplitude of the current is small, the interaction builds up near a number of its wavelengths until the train of surface waves is significantly modified. The equations governing the modifications are derived, using the method of multiple scales, and the roles of the Döppler shift and the radiation stress in resonant situations are elucidated. Three-dimensional interactions are discussed and an analogy is drawn between the fundamental equation describing the interactions and Schrödinger's equation.

In finding the resistance to the motion of a closely fitting plug along a tube filled with fluid, allowance must be made for the leakage of fluid through the gap between the plug and the tube. If there is no leakage, theoresistance is theoretically infinite. For a plug of length d in a tube of radius a and with a gap b between the plug and tube, the dominant term in the resistance comes from the flow in the gap, and is proportional to da/b. If, however, the plug is very short, so that it may be considered as a disk, the calculation of the flow near the disk shows that the resistance is now proportional to aln (a/b) and the presence of the disk increases the force on the tube by an amount equivalent to an increase in length of the tube by only a few radii.