Duality in Analytic Number TheoryCambridge University Press, 13 févr. 1997 In this stimulating book, aimed at researchers both established and budding, Peter 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: historical background is woven into the narrative, variant proofs illustrate obstructions, false steps and the development of insight, in a manner reminiscent of Euler. It is shown 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 until now beyond reach. Connections with other branches of analysis are illustrated by over 250 exercises, structured in chains about individual topics. |
Table des matières
| 1 | |
Background philosophy | 16 |
Operator norm inequalities | 18 |
Dual norm inequalities | 25 |
Including the Large Sieve | 32 |
Deriving the approximate functional equations | 48 |
Solving the approximate functional equations | 52 |
Almost linear Almost exponential | 68 |
The operator T on L 2 | 159 |
The operator T on L α and other spaces | 169 |
The operator D and differentiation The operator T and the convergence of measures | 183 |
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 |
Additive functions of class ℒα A first application of the method | 79 |
First Approach | 84 |
Second Approach | 93 |
Third Approach | 101 |
Why the form? | 111 |
Theorems of Wirsing and Halász | 115 |
Again Wirsings Theorem | 122 |
The prime number theorem | 127 |
Finitely distributed additive functions | 133 |
Multiplicative functions of the class ℒα Mean value zero | 139 |
Including logarithmic weights | 148 |
Encounters with Ramanujans function τn | 151 |
Large moduli | 239 |
Maximal inequalities | 254 |
Shift operators and orthogonal duals | 271 |
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 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 estimate Euler products exercise finite Fourier analysis function f g belongs Halász Hölder's inequality inner product integers interval Large Sieve Lemma loga logloga Lºn Math mean value zero method mod q Moreover multiplicative functions non-zero norm notation nsac obtain operator polynomial positive integer powers q prime number theorem prime powers proof of Lemma proof of Theorem Prove Ramanujan Re(s replaced representation residue classes result satisfy self-adjoint series XD space stable dual sufficiently large summation Theorem 9.1 Turán-Kubilius inequality uniformly bounded unit disc upper bound valid vector XC g(n