Assuming the Riemann Hypothesis, Montgomery showed by means of his paircorrelation method that at least two-thirds of the zeros of Riemann'szeta-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 atleast 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 andonly if it isdihedral, 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 solveseveral relative class number one problems. (1) We solve the relative class number one problem for thedihedral CM-fields of 2-power degrees. First, we remind the reader of the characterization of the dihedralCM-fields of 2-power degrees with odd relative class numbers. Second, we give lower bounds on relative classnumbers of dihedral CM-fields of 2-power degrees with odd relative class numbers. We thus obtain an upper boundon 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-fourdihedral CM-fields of 2-power degrees with relativeclass numbers equal to 1, and show that exactly twenty-oneof them have class number 1. (2) We determine all the non-abelian normal CM-fields of degree 16 with Galoisgroup $G_9$ which have relative class number 1 (there is only one such number field), and then those whichhave class number 1 (there is only one such number field). (3) We determine some of the non-abelian normalCM-fields $N =N_1N_2$ of degree 16 which are composita of two normal octic CM-fields $N_1$ and $N_2$ with thesame 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 onlyone such compositum with relative class number 1 and that this compositum has class number 1.

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

A Mackey functor $M$ is a structure analogous tothe representation ring functor $H \mapsto R(H)$ encoding good formal behaviour under induction andrestriction. More explicitly,$M$ associates an abelian group $M(H)$ to each closed subgroup $H$ of a fixedcompact Lie group $G$, and to each inclusion $K \subseteq H$ it associates a restriction map ${\rmres}^H_K:M(H) \rightarrow M(K)$ and an induction map${\rm ind}^H_K:M(K) \rightarrow M(H)$. This paper gives ananalysis of the category of Mackey functors $M$ whose values are rational vector spaces: such a Mackey functormay 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 rationalMackey functors is equivalent to the category of rational continuousWeyl-toral modules. In Part II this willbe used to give an algebraic analysis ofthe category of rational Mackey functors, showing in particular thatit has homological dimension equal to the rank of the group.

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

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

1991Mathematics Subject Classification: 35P20.

In this paper Hill's equation $y''+qy=Ey$, where $q$ is a complex-valued function with inverse squaresingularities, 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 asymptoticdistribution of (semi-)periodic and Sturm-Liouville eigenvalues are obtained. It is proved that a certainsubset of the set of Floquet solutions is a line bundle on a certain analytic curve in ${\Bbb C}^2$. Weestablish necessary and sufficient conditions for $q$ to be algebro-geometric, that is, to be a stationarysolution of some equation in the Korteweg-de Vries (KdV) hierarchy. To do this a distinction between movableand immovable Dirichlet eigenvalues is employed. Finally, an example showing that the finite-band property doesnot imply that $q$ is algebro-geometric is given. This is in contrast to the case where $q$ is real andnon-singular.

1991 Mathematics Subject Classification: 34L40, 14H60.

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

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

In this paper exact asymptotic formulae are found for singular values of the Cauchyoperator and the logarithmic potential type operator (on a bounded domain), as well as their productswith Bergman's projection. It is shown that these spectral characteristics detect geometric propertiesof a domain $\Omega$ (area and the length of the boundary). The hypothesis “can we hear theshape of a drum”, from a paper by J.M. Anderson, D. Khavinson, and V. Lomonosov [‘Spectralpropertiesof 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 SubjectClassification: 47B10.

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

1991 Mathematics Subject Classification: 57N13.

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

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