Duality in Analytic Number Theory

Couverture
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

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Expressions et termes fréquents

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.

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