Duality in Analytic Number Theory
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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Including the Large Sieve
Deriving the approximate
Almost linear Almost exponential
Theorems of Wirsing and Halasz
Again Wirsings Theorem
Multiplicative functions on arithmetic progressions Wiener phenomenon
Fractional power Large Sieves Operators involving primes
Probability seen from number theory
Shift operators and orthogonal duals
Differences of additive functions Local inequalities
The prime number theorem
Finitely distributed additive functions
Multiplicative functions of the class Ca Mean value zero
Including logarithmic weights
Encounters with Ramanujans function tti
The operator T on L2
The operator T on La and other spaces
The operator D and differentiation The operator T and the convergence of measures
Towards the discrete derivative
Autres éditions - Tout afficher
additive functions analogue analytic number theory application of Holder's approximate functional equations argument arithmetic function asserts asymptotic bounded uniformly Cauchy-Schwarz inequality Chapter 20 complex numbers complex unit disc complex valued conjecture convergence defined denote Dirichlet characters Dirichlet series duality eigenvalue eigenvector employ established estimate Euler products exceeding exercise factor finite fixed Fourier analysis function g g belongs generalisation given Haar measure Halasz holds uniformly hypothesis inner product integers interval La(Cs Large Sieve Lemma Let g log log logx mean value zero method mod q moduli q Moreover multiplicative function non-negative non-zero norm notation number of prime obtain operator particular polynomial positive integer powers q prime number theorem prime powers proof of Lemma proof of Theorem Prove replaced representation result satisfy self-adjoint stable dual sufficiently large summation Theorem 9.1 Turan-Kubilius inequality uniformly bounded unit disc upper bound valid vector
Page 328 - On the estimation of the second central moment for strongly additive arithmetic functions.
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.