Assuming the Riemann Hypothesis, Montgomery showed by means of his pair correlation method that at least two-thirds of the zeros of Riemann's zeta-function are simple. Later he and Taylor improved this to 67.25 percent and, more recently, Cheer and Goldston increased the percentage to 67.2753. Here we prove by a new method that if the Riemann and Generalized Lindel\"of Hypotheses hold, then at least 70.3704 percent of the zeros are simple and at least 84.5679 percent are distinct. Our method uses mean value estimates for various functions defined by Dirichlet series sampled at the zeros of the Riemann zeta-function.

1991 Mathematics Subject Classification: 11M26.

First, we prove that a non-abelian normal CM-field of degree 16 has odd relative class number if and only if it is dihedral, or is a compositum of two normal octic CM-fields with the same maximal real subfield, or has Galois group $G_9 =\langle a,b,z: a^2=b^2=z^4=1,b^{-1}ab=az^2,az=za,bz=zb\rangle$. Then, we solve several relative class number one problems. (1) We solve the relative class number one problem for the dihedral CM-fields of 2-power degrees. First, we remind the reader of the characterization of the dihedral CM-fields of 2-power degrees with odd relative class numbers. Second, we give lower bounds on relative class numbers of dihedral CM-fields of 2-power degrees with odd relative class numbers. We thus obtain an upper bound on the discriminants of the dihedral CM-fields of 2-power degrees with relative class number equal to 1. Third, we compute the relative class numbers of all the dihedral CM-fields of 2-power degrees with odd relative class numbers and discriminants less than or equal to this latter bound. We end up with a list of twenty-four dihedral CM-fields of 2-power degrees with relativeclass numbers equal to 1, and show that exactly twenty-one of them have class number 1. (2) We determine all the non-abelian normal CM-fields of degree 16 with Galois group $G_9$ which have relative class number 1 (there is only one such number field), and then those which have class number 1 (there is only one such number field). (3) We determine some of the non-abelian normal CM-fields $N =N_1N_2$ of degree 16 which are composita of two normal octic CM-fields $N_1$ and $N_2$ with the same maximal real subfield which have relative class number 1, and then those which have class number 1. Indeed, we focus on the case where one of the $N_i$ is a quaternion octic CM-field and prove that there is only one such compositum with relative class number 1 and that this compositum has class number 1.

1991 Mathematics Subject Classification: primary 11R29; secondary 11R21, 11R42, 11M20, 11Y40.

A Mackey functor $M$ is a structure analogous to the representation ring functor $H \mapsto R(H)$ encoding good formal behaviour under induction and restriction. More explicitly, $M$ associates an abelian group $M(H)$ to each closed subgroup $H$ of a fixed compact Lie group $G$, and to each inclusion $K \subseteq H$ it associates a restriction map ${\rm res}^H_K:M(H) \rightarrow M(K)$ and an induction map${\rm ind}^H_K:M(K) \rightarrow M(H)$. This paper gives an analysis of the category of Mackey functors $M$ whose values are rational vector spaces: such a Mackey functor may be specified by giving a suitably continuous family consisting of a ${\Bbb Q} \pi_0(W_G(H))$-module $V(H)$ for each closed subgroup $H$ with restriction maps $V(\hat{K}) \rightarrow V(K)$ whenever $K$ is normal in $\hat{K}$ and $\hat{K}/K$ is a torus (a ‘continuous Weyl-toral module’). We show that the category of rational Mackey functors is equivalent to the category of rational continuous Weyl-toral modules. In Part II this will be used to give an algebraic analysis of the category of rational Mackey functors, showing in particular that it has homological dimension equal to the rank of the group.

1991 Mathematics Subject Classification: 19A22, 20C99, 22E15, 55N91, 55P42, 55P91.

The leading asymptotic term for the function that counts the eigenvalues of the Stokes operator is determined for fairly general underlying bounded domains. Moreover, the remainder is estimated in terms of the fractality of the boundary of the domain. The results obtained resemble corresponding ones for the Dirichlet Laplacian.

1991 Mathematics Subject Classification: 35P20.

In this paper Hill's equation $y''+qy=Ey$, where $q$ is a complex-valued function with inverse square singularities, is studied. Results on the dependence of solutions to initial value problems on the parameter $E$ and the initial point $x_0$, on the structure of the conditional stability set, and on the asymptotic distribution of (semi-)periodic and Sturm-Liouville eigenvalues are obtained. It is proved that a certain subset of the set of Floquet solutions is a line bundle on a certain analytic curve in ${\Bbb C}^2$. We establish necessary and sufficient conditions for $q$ to be algebro-geometric, that is, to be a stationary solution of some equation in the Korteweg-de Vries (KdV) hierarchy. To do this a distinction between movable and immovable Dirichlet eigenvalues is employed. Finally, an example showing that the finite-band property does not imply that $q$ is algebro-geometric is given. This is in contrast to the case where $q$ is real and non-singular.

1991 Mathematics Subject Classification: 34L40, 14H60.

We give a complete classification (up to unitary equivalence) of extensions of $C(X)$ by a separable simple AF-algebra $A$ with a unique trace (up to scalar multiples), where $X$ is a compact subset of the plane. In particular, we show that there are non-trivial extensions $\tau$ such that $[\tau]=0$ in ${\rm Ext}(C(X),A).$ A new index is introduced to determine when an extension is trivial. Extensions of $C(S^2)$ and other algebras are also studied. Our results work for a larger class of C*-algebras of real rank zero.

1991 Mathematics Subject Classification: primary 46L05,46L35; secondary 46L80.

In this paper exact asymptotic formulae are found for singular values of the Cauchy operator and the logarithmic potential type operator (on a bounded domain), as well as their products with Bergman's projection. It is shown that these spectral characteristics detect geometric properties of a domain $\Omega$ (area and the length of the boundary). The hypothesis “can we hear the shape of a drum”, from a paper by J.M. Anderson, D. Khavinson, and V. Lomonosov [‘Spectral propertiesof some integral operators arising in potential theory’, {\em Quart.\ J. Math.\ Oxford} (2) 43 (1992) 387-407], is correct in the above sense.

1991 Mathematics Subject Classification: 47B10.

We determine strong constraints on the generalized Euler invariants of Seifert bundles over non-orientable base orbifolds which may embed as topologically locally flat submanifolds of $S^4$. In particular, a circle bundle over a non-orientable surface $F$ embeds if and only if it embeds as the boundary of a regular neighbourhood of an embedding of $F$ in $S^4$, and we show that precisely thirteen geometric 3-manifolds with elementary amenable fundamental groups embed. With the exception of the Poincaré homology sphere, each member of the latter class may be obtained by 0-framed surgery on a link which is the union of two slice links, and so embeds smoothly in $S^4$.

1991 Mathematics Subject Classification: 57N13.

This paper concerns a Markov operator $T$ on a space $L_1$, and a Markov process $P$ which defines a Markov operator on a space $M$ of finite signed measures. For $T$, the paper presents necessary and sufficient conditions for: \begin{enumerate}\item [(a)] the existence of invariant probability densities (IPDs)\item [(b)] the existence of strictly positive IPDs, and\item [(c)] the existence and uniqueness of IPDs.\end{enumerate} Similar results on invariant probability measures for $P$ are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

1991 Mathematics Subject Classification: 60J05, 47B65, 47N30.