Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

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Analytic Number Theory

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and Kirchberg's factorization property Quasidiagonal C*-algebras AF embeddablity Local reflexivity and other tensor product conditions Summary and open problems Special topics: Simple $\textrm{C}^*$-algebras Approximation properties for groups Weak expectation property and local lifting property Weakly exact von Neumann algebras Applications: Classification of group von Neumann algebras Herrero's approximation problem Counterexamples in $\textrm{K}$-homology and $\textrm{K}$-theory Appendices: Ultrafilters and ultraproducts Operator spaces, completely bounded maps and duality Lifting theorems Positive definite functions, cocycles and Schoenberg's Theorem Groups and graphs Bimodules over von Neumann algebras Bibliography Notation index Subject index.

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C*-Algebras and Finite-Dimensional Approximations

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

I shall emphasize the role of automorphic forms in harmonic analysis because they are indispensable in analytic number theory, which is my primary subject of interest. A lot has been presented to the general forum about modular forms in algebraic number theory, in particular after resolution of the Fermat last theorem [W], [BCDT]. Therefore I shall limit my venture to the topics which help to grasp the essence of modularity in number theory as a whole.

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Automorphic Number Theory

Introduction. Number theory and modular forms have been intimately related from the very beginnings of the latter. Very well known are connections with higher arithmetic and algebraic number theory. In the last decade there have been considerable developments in modular forms in relation to analytic number theory. Its foundation rests on the spectral theory of automorphic functions due to H. Maass [26] and A. Selberg [34]. Initially the spectral theory served to give extremely sharp estimates for certain exponential sums of primary importance for problems in analytic number theory. This area of research was given the nickname "Kloostermania." Today analytic number theory is paying back. The connection in question has been reversed to that the machinery of analytic number theory is working to improve the spectral theory of automorphic functions, at least in the most interesting case of congruence groups. This indispensible interrelation is the subject of my talk. We shall also mention some applications to particular problems. My favorite one is the mean-value theorem for primes in arithmetic progressions to large moduli. In 1965 E. Bombieri [1] and A. I. Vinogradov [40] proved powerful results which serve as a substitute for the Riemann hypothesis for Dirichlet's L-series, especially in reference to sieve methods. In a series of joint papers with E. Bombieri, E. Fouvry, and J. Friedlander (see the references in [2]) we attempted to extend the Bombieri and Vinogradov results beyond the capability of the Riemann hypothesis. The dream just came through in [2] where we prove that for any a ^ 0, A > 0, and x > 3 one has,

Spectral Theory of Automorphic Functions and Recent Developments in Analytic Number Theory

Exceptional zeros and the Goldbach problem

We survey some of the history and results related to the topic of the title with an emphasis admittedly biased toward our joint works thereon.

Henryk Iwaniec

Papers

Automorphic Number Theory

Spectral Theory of Automorphic Functions and Recent Developments in Analytic Number Theory

Exceptional zeros and the Goldbach problem

The Temptation of the Exceptional Characters

The Dirichlet polynomials