Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

Part II. Specific Universal Families and Hypercyclic Operators 3. The real analysis setting 3a. Universal power and Taylor series 3b. Universal primitives 3c. Universal orthogonal series 3d. Universal series for convergence a.e. 3e. Further real universalities and hypercyclicities 4. The complex analysis setting 4a. Universal and hypercyclic composition operators 4b. Holomorphic monsters 4c. Hypercyclic differential operators 4d. Universal power and Taylor series 4e. Universal matrices 5. Hypercyclic operators in classical Banach spaces 6. A concrete universal object: the Riemann zeta-function

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Universal families and hypercyclic operators

Preface 1 Elements of the probability theory 2 Dirichlet series and Dirichlet polynomials 3 Limit theorems for the modulus of the Riemann Zeta-function 4 Limit theorems for the Riemann Zeta-function on the complex plane 5 Limit theorems for the Riemann Zeta-function in the space of analytic functions 6 Universality theorem for the Riemann Zeta-function 7 Limit theorem for the Riemann Zeta-function in the space of continuous functions 8 Limit theorems for Dirichlet L-functions 9 Limit theorem for the Dirichlet series with multiplicative coefficients References Notation Subject index

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Limit Theorems for the Riemann Zeta-Function

This paper studies the Lerch zeta-function L(λ αs) which was introduced by M. Lerch in 1887. The case of non-integer parameter λ is considered, and then L(λ αs) is an entire function. First, the universality of this function with transcendental parameter a is obtained. From this the functional independence is derived. The third part of the paper is devoted to the zero-distribution of the Lerch zeta-function. Zero-free regions are indicated on the right as well on the left of the imaginary axis when λ ≠ 1/2. If λ = 1/2, then zero-free regions lie in the left half-plane. Finally, estimates for the number of zeros in the critical strip are given.

The Lerch zeta-function

The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

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The joint universality and the functional independence for Lerch zeta-functions

This remarkable result raised much attention among specialists, and Reich [18], [19], Gonek [4], Good [5], Bagchi [1], [2], and the first author [8]-[12], [14] improved and generalized the Voronin theorem to various other Dirichlet series including Dirichlet L , and Dedekind, Hurwitz, and Lerch zeta-functions. It is the purpose of the present paper to prove the universality theorem for zetafunctions attached to certain cusp forms. Let F (z) be a holomorphic cusp form of weight κ for the full modular group SL(2, Z) , and assume that F (z) is a normalized eigenform. Then F (z) has the Fourier series expansion

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The universality of zeta-functions attached to certain cusp forms

The first part of the paper contains a survey on the universality of zeta-functions. Zeta-functions with Euler's product as well as zeta-functions without Euler's product are discussed. Also, the joint universality theorems are considered. In the second part of the paper the universality of zeta-functions of finite Abelian groups of rank ≤3 is proved.

Antanas Laurinčikas

Papers

Limit Theorems for the Riemann Zeta-Function

The Lerch zeta-function

The joint universality and the functional independence for Lerch zeta-functions

The universality of zeta-functions attached to certain cusp forms

The Universality of Zeta-Functions