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By Peter D. T. A. Elliott
Cambridge Tracts in Mathematics
(No. 122)
Publisher: Cambridge University Press
Print Publication Year:1997
Online Publication Date:November 2011
Online ISBN:9780511983405
Hardback ISBN:9780521560887
Paperback ISBN:9780521058087
Book DOI: http://dx.doi.org/10.1017/CBO9780511983405
Subjects: Number theory , Discrete Mathematics, Information Theory and Coding
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.
Reviews:
pp. i-iv
pp. v-vi
pp. vii-viii
pp. ix-ix
pp. x-xii
pp. xiii-xviii
0 - Duality and Fourier analysis: Read PDF
pp. 1-15
1 - Background philosophy: Read PDF
pp. 16-17
2 - Operator norm inequalities: Read PDF
pp. 18-24
3 - Dual norm inequalities: Read PDF
pp. 25-31
4 - Exercises: Including the Large Sieve: Read PDF
pp. 32-47
5 - The method of the stable dual (1): Deriving the approximate functional equations: Read PDF
pp. 48-51
6 - The method of the stable dual (2): Solving the approximate functional equations: Read PDF
pp. 52-67
7 - Exercises: Almost linear, Almost exponential: Read PDF
pp. 68-78
8 - Additive functions of class ℒα. A first application of the method: Read PDF
pp. 79-83
9 - Multiplicative functions of the class ℒα: First Approach: Read PDF
pp. 84-92
10 - Multiplicative functions of the class ℒα: Second Approach: Read PDF
pp. 93-100
11 - Multiplicative functions of the class ℒα: Third Approach: Read PDF
pp. 101-110
12 - Exercises: Why the form?: Read PDF
pp. 111-114
13 - Theorems of Wirsing and Halász: Read PDF
pp. 115-121
14 - Again Wirsing's Theorem: Read PDF
pp. 122-126
15 - Exercises: The prime number theorem: Read PDF
pp. 127-132
16 - Finitely distributed additive functions: Read PDF
pp. 133-138
17 - Multiplicative functions of the class ℒα. Mean value zero: Read PDF
pp. 139-147
18 - Exercises: Including logarithmic weights: Read PDF
pp. 148-150
19 - Encounters with Ramanujan's function τ(n): Read PDF
pp. 151-158
20 - The operator T on L2 : Read PDF
pp. 159-168
21 - The operator T on Lα and other spaces: Read PDF
pp. 169-182
22 - Exercises: The operator D and differentiation. The operator T and the convergence of measures: Read PDF
pp. 183-189
23 - Pause: Towards the discrete derivative: Read PDF
pp. 190-204
24 - Exercises: Multiplicative functions on arithmetic progressions. Wiener phenomenon: Read PDF
pp. 205-210
25 - Fractional power Large Sieves. Operators involving primes: Read PDF
pp. 211-231
26 - Exercises: Probability seen from number theory: Read PDF
pp. 232-234
27 - Additive functions on arithmetic progressions: Small moduli: Read PDF
pp. 235-238
28 - Additive functions on arithmetic progressions: Large moduli: Read PDF
pp. 239-253
29 - Exercises: Maximal inequalities: Read PDF
pp. 254-270
30 - Shift operators and orthogonal duals: Read PDF
pp. 271-274
31 - Differences of additive functions. Local inequalities: Read PDF
pp. 275-284
32 - Linear forms in shifted additive functions: Read PDF
pp. 285-294
33 - Exercises: Stability. Correlations of multiplicative functions: Read PDF
pp. 295-301
34 - Further readings: Read PDF
pp. 302-319
35 - Rückblick (after the manner of Johannes Brahms): Read PDF
pp. 320-320
pp. 321-332
pp. 333-334
pp. 335-341
No references available.