An oversight by the typesetters has led to the omission of the summation condition on the second summation of the first displayed equation on page 392 of [1]. The proof of Lemma 4.1 is consequently very difficult to follow. The expression for Ip(α) should read

In addition, we take this opportunity to note that on page 391, in the two displayed equations following equation (3.15), the summations involving ud+1 should read

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

Let s1, s2, … denote the squarefree numbers in ascending order. In [1], Erdős showed that, if 0 ≤ γ ≤ 2, then

where B(γ) is a function only of γ. In 1973 Hooley [4] improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for

in contrast to the range

derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.

Let

and A1, A2, …, An, a1, a2, …, an be positive integers. We consider the related integrals

and

Let γ:x = x(u) for a≤ u ≤ b be a closed curve in n dimensional euclidean space En (for n ≥ 2) referred to some point P0, which does not lie on γ, as origin. We suppose that γ is smooth in the sense that the cartesian coordinates are class C2 functions of the parameter u, and that dx/du is non-vanishing so that a tangent vector is everywhere well defined. These properties are also assumed to hold in an obvious way at the join of the end points u = a and u = b. As P moves on γ its position vector x(u) intersects the surface of the unit sphere centred at P0 in a closed curve γ0. Note that γ0: x = x0(u) may not be smooth everywhere. If there are points P on γ where P0P is tangential to γ at P then dx0/du =0 at the corresponding point on γ0, and γ0 may have a cusp there. We assume that γ0 is smooth except at a finite number of points. We define the total swing of the position vector to γ to be the arc length L0 of γ0. Clearly L0 is not an invariant but depends on the choice of the origin P0 in relation to γ. The total (first) curvature of γ is an invariant and is defined by

where s is the arc length on γ, K(S) is the curvature and ‖v‖ is the euclidean norm of the vector v. Note that LT is also the arc length of the closed curve γ1, described on a unit sphere by the unit tangent t = dx/ds to γ as position vector, with the centre of the sphere as origin, γ1 is the spherical indicatrix of t. Our purpose is to establish the result

for all choices of P0.

An oncoming two-dimensional laminar boundary layer that develops an unstable inflection point and becomes three-dimensional is described by the Hall-Smith (1991) vortex/wave interaction equations. These equations are now examined in the neighbourhood of the position where the critical surface starts to form. A consistent structure is established in which an inviscid core flow is matched to a viscous buffer-layer solution where the appropriate jump condition on the transverse shear stress is satisfied. The final result is a bifurcation equation for the (constant) amplitude of the wave pressure. A representative classical velocity profile is considered to illustrate solutions of this equation for a range of values of the wave-numbers.

Suppose that is a distribution of N points in U0, the closed disc of unit area and centred at the origin 0. For every measurable set B in ℝ2, let Z[; B] denote the number of ponts of in B, and write

where µ denotes the usual measure in ℝ2