# Duality in Analytic Number Theory

Cambridge University Press, 13 févr. 1997 - 341 pages
In this stimulating book, Elliott demonstrates a method and a motivating philosophy that combine to cohere a large part of analytic number theory, including the hitherto nebulous study of arithmetic functions. Besides its application, the book also illustrates a way of thinking mathematically: The author weaves historical background into the narrative, while variant proofs illustrate obstructions, false steps and the development of insight in a manner reminiscent of Euler. He demonstrates how to formulate theorems as well as how to construct their proofs. Elementary notions from functional analysis, Fourier analysis, functional equations, and stability in mechanics are controlled by a geometric view and synthesized to provide an arithmetical analogue of classical harmonic analysis that is powerful enough to establish arithmetic propositions previously beyond reach. Connections with other branches of analysis are illustrated by over 250 exercises, topically arranged.

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### Table des matières

 Background philosophy 16 Including the Large Sieve 32 Deriving the approximate 48 Almost linear Almost exponential 68 First Approach 84 Third Approach 101 Theorems of Wirsing and Halasz 115 Again Wirsings Theorem 122
 Multiplicative functions on arithmetic progressions Wiener phenomenon 205 Fractional power Large Sieves Operators involving primes 211 Probability seen from number theory 232 Small moduli 235 Large moduli 239 Maximal inequalities 254 Shift operators and orthogonal duals 271 Differences of additive functions Local inequalities 275

 The prime number theorem 127 Finitely distributed additive functions 133 Multiplicative functions of the class Ca Mean value zero 139 Including logarithmic weights 148 Encounters with Ramanujans function tti 151 The operator T on L2 159 The operator T on La and other spaces 169 The operator D and differentiation The operator T and the convergence of measures 183 Towards the discrete derivative 190
 Linear forms in shifted additive functions 285 Stability Correlations of multiplicative functions 295 Further readings 302 Rtickblick after the manner of Johannes Brahms 320 References 321 Author index 333 Subject index 335 Droits d'auteur

### Fréquemment cités

Page 328 - On the estimation of the second central moment for strongly additive arithmetic functions.
Page 326 - P. On the distribution function of additive functions, Ann. of Math. 47 (1946), 1-20.
Page 326 - Multiplicative functions on arithmetic progressions VI: More middle moduli, J. Number Theory 44 (2) (1993), 178-208.
Page 322 - Quelques proprietes des fonctions multiplicatives de module au plus egal a 1, CR Acad. Sci. Paris Ser.
Page 326 - Elliott, PDTA and Halberstam, H. A conjecture in prime number theory, Symposia Mathematica, IV, Academic Press, London and New York, 1970, 59-72.
Page 326 - Erdos, P., Kac, M. , On the Gaussian law of errors in the theory of additive functions, Amer. J. Math.
Page 328 - Improvement of the estimation of the second central moment for additive arithmetical functions, Liet.