Analytic Number Theory: Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 11-18, 2002 (Lecture Notes in Mathematics / Fondazione C.I.M.E., Firenze)

The four contributions collected in this volume deal with several advanced results in analytic number theory. Friedlander’s paper contains some recent achievements of sieve theory leading to asymptotic formulae for the number of primes represented by suitable polynomials. Heath-Brown's lecture notes mainly deal with counting integer solutions to Diophantine equations, using among other tools several results from algebraic geometry and from the geometry of numbers. Iwaniec’s paper gives a broad picture of the theory of Siegel’s zeros and of exceptional characters of L-functions, and gives a new proof of Linnik’s theorem on the least prime in an arithmetic progression. Kaczorowski’s article presents an up-to-date survey of the axiomatic theory of L-functions introduced by Selberg, with a detailed exposition of several recent results. Contains A Representative Collection Of Surveys By Experts In Mathematical Relativity Writing About The Status Of, And Problems In, Their Fields. This Book Includes Four Contributions For Each Of The Mathematical Areas: Differential Geometry And Differential Topology, Analytical Methods And Differential Equations, And Numerical Methods. A Personal Perspective On Global Lorentzian Geometry / P.e. Ehrlich -- The Space Of Null Geodesics (and A New Causal Boundary) / R.j. Low -- Some Variational Problems In Semi-riemannian Geometry / A. Masiello -- On The Geometry Of Pp-wave Type Spacetimes / J.l. Flores And M. Sánchez -- Concepts Of Hyperbolicity And Relativistic Continuum Mechanics / R. Beig -- Elliptic Systems / S. Dain -- Mathematical Properties Of Cosmological Models With Accelerated Expansion / A.d. Rendall -- The Poincaré Structure And The Centre-of-mass Of Asymptotically Flat Spacetimes / L.b. Szabados -- Computer Simulation : A Tool For Mathematical Relativity - And Vice Versa / B.k. Berger -- On Boundary Conditions For The Einstein Equations / S. Fritelli And R. Gómez -- Recent Analytical And Numerical Techniques Applied To The Einstein Equations / D. Neilsen ... [et Al.] -- Some Mathematical Problems In Numerical Relativity / M. Babiuc, B. Szilágyi, J. Winicour. Jörg Frauendiener, Domenico J.w. Giulini, Volker Perlick (eds.) ; With A Foreword By Roger Penrose. Includes Bibliographical References And Index. Also Available On The Internet Via The World Wide Web. The general theory of relativity, as formulated by Albert Einstein in 1915, provided an astoundingly original perspective on the physical nature of gr- itation, showing that it could be understood as a feature of a curvature in the four-dimensional continuum of space-time. Now, some 90 years later, this extraordinary theory stands in superb agreement with observation, prov- ing a profound accord between the theory and the actual physical behavior of astronomical bodies, which sometimes attains a phenomenal precision (in one case to about one part in one hundred million million, where several d- ferent non-Newtonian e?ects, including the emission of gravitational waves, are convincingly con?rmed). Einstein's tentative introduction, in 1917, of an additional term in his equations, speci?ed by a “cosmological constant”, - pearsnowtobeobservationallydemanded,andwiththistermincluded,there is no discrepancy known between Einstein's theory and classical dynamical behavior, from meteors to matter distributions at the largest cosmological scales. One of Einstein's famous theoretical predictions that light is bent in a gravitational?eld (which had been only roughly con?rmed by Eddington's solareclipsemeasurementsattheIslandofPrincipein1919,butwhichisnow very well established) has become an important tool in observational cosm- ogy, where gravitational lensing now provides a unique and direct means of measuring the mass of very distant objects. General relativity ranks among the most accurately tested fundamental theories in all of physics. Deficiencies in mathematical and conceptual understanding still exist, hampering further progress. This book collects surveys by experts in mathematical relativity writing about the current status of, and problems in, their fields. There are four contributions for each of the following mathematical areas: differential geometry and differential topology, analytical methods and differential equations, and numerical methods. The Four Papers Collected In This Book Discuss Advanced Results In Analytic Number Theory, Including Recent Achievements Of Sieve Theory Leading To Asymptotic Formulae For The Number Of Primes Represented By Suitable Polynomials; Counting Integer Solutions To Diophantine Equations, Using Results From Algebraic Geometry And The Geometry Of Numbers; The Theory Of Siegel’s Zeros And Of Exceptional Characters Of L-functions; And An Up-to-date Survey Of The Axiomatic Theory Of L-functions Introduced By Selberg. "This book contains a representative collection of surveys by experts in mathematical relativity writing about the current status of, and problems in, their fields. There are four contributions for each of the following mathematical areas: differential geometry and differential topology, analytical methods and differential equations, and numerical methods. This book addresses graduate students and specialist researchers alike."--Jacket Producing Prime Numbers Via Sieve Methods / John B. Friedlander -- Counting Rational Points On Algebraic Varieties / D.r. Heath-brown -- Conversations On The Exceptional Character / Henryk Iwaniec -- Axiomatic Theory Of L-functions: The Selberg Class / Jerzy Kaczorowski. J.b. Friedlander ... [et Al.] ; Editors: A. Perelli, C. Viola. Includes Bibliographical References. Front Matter....Pages I-XI Producing Prime Numbers via Sieve Methods....Pages 1-49 Counting Rational Points on Algebraic Varieties....Pages 51-95 Conversations on the Exceptional Character....Pages 97-132 Axiomatic Theory of L -Functions: the Selberg Class....Pages 133-209 Back Matter....Pages 211-216