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

One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).

https://www.ems-ph.org/journals/show_pdf.php?issn=0213-2230&vol=11&iss=2&rank=8

On the Sphere Problem

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$, where $\psi$ is a character of the ideal class group of the imaginary quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text {squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the vicinity of the central point $s = 1/2$, the average distribution of these zeros (for $D\longrightarrow \infty$) is governed by the symplectic distribution. By averaging over $D$, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve techniques are applied.

Low-lying zeros of dihedral L-functions

We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> \sqrt q (log q)^{-A} for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-function of the number field and so contains the Riemann zeta-function as a factor. Thus, as a corollary to our main result, we prove that this lower bound for h follows from the hypothesis that there are sufficiently many pairs of adjacent zeros of the Riemann zeta-function on the critical line whose spacing is slightly smaller than 1/2 of the average spacing.

Henryk Iwaniec

Papers

On the Sphere Problem

Low-lying zeros of dihedral L-functions

Spacing of zeros of Hecke L-functions and the class number problem

The divisor problem for arithmetic progressions

On zeros of Dirichlet'sL series