Duality in Analytic Number TheoryCambridge 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. |
Table des matières
Duality and Fourier analysis | 1 |
Background philosophy | 16 |
Including the Large Sieve | 32 |
Deriving the approximate | 48 |
Almost linear Almost exponential | 68 |
First Approach 84 4 | 84 |
Third Approach | 101 |
Theorems of Wirsing and Halász | 115 |
Towards the discrete derivative | 190 |
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 |
Again Wirsings Theorem | 122 |
The prime number theorem | 127 |
Finitely distributed additive functions | 133 |
Multiplicative functions of the class La Mean value zero | 139 |
Including logarithmic weights | 148 |
Encounters with Ramanujans function 7N | 151 |
The operator T on L² | 167 |
The operator T on Lª and other spaces | 169 |
The operator D and differentiation The operator T and the convergence of measures | 183 |
Differences of additive functions Local inequalities | 275 |
Linear forms in shifted additive functions | 285 |
Stability Correlations of multiplicative functions | 295 |
Further readings | 302 |
Rückblick after the manner of Johannes Brahms | 320 |
| 321 | |
| 333 | |
| 335 | |
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Expressions et termes fréquents
additive arithmetic functions additive functions analogue analytic number theory application of Hölder's approximate functional equations argument arithmetic functions asserts asymptotic bounded uniformly Cauchy-Schwarz inequality Chapter complex numbers complex unit disc condition conjecture convergence defined denote Dirichlet characters Dirichlet series duality eigenvalue eigenvector employ established estimate Euler products exceeding exercise finite Fourier analysis function f g belongs generalisation given Halász Hölder's inequality inner product integers interval Large Sieve Lemma log log Math mean value zero method mod q Moreover multiplicative functions n=r(mod non-zero norm notation number of prime obtain operator polynomial positive integer powers q prime number theorem prime powers proof of Lemma proof of Theorem Prove Ramanujan replaced representation residue classes result satisfy self-adjoint space stable dual sufficiently large summation Theorem 9.1 Turán-Kubilius inequality uniformly bounded unit disc upper bound valid vector x(log Σ Σ ΣΣ